A geometrical theory of gliding motility based on cell shape and surface flow

Significance

The four main motility modes of biological cells are swimming, crawling, twitching, and gliding. Gliding is typically used by cells that achieve high speeds in a solid environment and often results from actively driven surface flow of adhesins. Here, we show theoretically that such surface flow typically leads to rotations of the cell around its long axis. Our theory suggests that the curved shapes of Plasmodium sporozoites and Toxoplasma tachyzoites as well as the internal tracks guiding surface flow of certain bacteria are evolutionary adaptations to avoid pure rotations and to generate productive forward movement. Our theory naturally explains why circular and helical cell trajectories are often observed for gliding motility and suggests design principles for synthetic microgliders.

Abstract

Gliding motility proceeds with little changes in cell shape and often results from actively driven surface flows of adhesins binding to the extracellular environment. It allows for fast movement over surfaces or through tissue, especially for the eukaryotic parasites from the phylum apicomplexa, which includes the causative agents of the widespread diseases malaria and toxoplasmosis. We have developed a fully three-dimensional active particle theory which connects the self-organized, actively driven surface flow over a fixed cell shape to the resulting global motility patterns. Our analytical solutions and numerical simulations show that straight motion without rotation is unstable for simple shapes and that straight cell shapes tend to lead to pure rotations. This suggests that the curved shapes of Plasmodium sporozoites and Toxoplasma tachyzoites are evolutionary adaptations to avoid rotations without translation. Gliding motility is also used by certain myxo- or flavobacteria, which predominantly move on flat external surfaces and with higher control of cell surface flow through internal tracks. We extend our theory for these cases. We again find a competition between rotation and translation and predict the effect of internal track geometry on overall forward speed. While specific mechanisms might vary across species, in general, our geometrical theory predicts and explains the rotational, circular, and helical trajectories which are commonly observed for microgliders. Our theory could also be used to design synthetic microgliders.

Life is motion on all scales, ranging from molecules through cells to organisms (1). Because cells are the smallest units of life, cell motility is a key aspect of many biological processes. While for free-living single organisms like bacteria or slime molds, this typically includes movement toward nutrients and formation of communities, for animal cells this includes processes such as development, immune response, regeneration, and metastasis (2). The relevance of cell motility for many different biological systems is mirrored by the large variety of physical mechanisms that are used to accomplish it (3). In a fluid environment, motility is often achieved by swimming with rotating or beating flagella. For animal cells, the most common form of motility is crawling, when the front is pushed forward by a lamellipodium and the back is pulled forward by actomyosin contractility (4). While swimming is typically as fast as 100 $\mathrm{\mu }$m/s, crawling is much slower, with a typical speed of 1 $\mathrm{\mu }$m/min.

Besides swimming and crawling, other major modes of cell motility are twitching and gliding, which both do not require major shape changes. While twitching uses pilli that extend and retract, gliding is usually based on surface flow of adhesion molecules; as these adhesins bind to the substrate, the surface flow is converted into forward movement of the cell. Because the internal structure of gliders does not need to change, high speeds are possible, typically in the range of 1 $\mathrm{\mu }$m/s. Gliding motility has evolved mainly for organisms that have to quickly move in a solid environment that provides substrates to which they can adhere. The model presented here is motivated by the well-conserved gliding machinery of apicomplexan parasites (5–7). The two most prominent examples are the genus of Plasmodium, which is the causative agent of malaria, and the species Toxoplasma gondii, causing toxoplasmosis (8).

Fig. 1A depicts the main elements in gliding motility of Plasmodium sporozoites (6, 9), which are the form of the malaria parasite that invades vertebrate hosts. Sporozoites have the shape of a 10 $\mathrm{\mu }$m long, thin, curved rod and move with high speeds up to 4 $\mathrm{\mu }$m/s to cross tissue barriers and escape the immune system of the host. Adhesins like the thrombospondin-related anonymous protein (10) are released at the front and effectively pulled toward the rear by an actomyosin system located below the membrane. On a flat (2D) substrate sporozoites primarily run on circular trajectories (9) (Fig. 1B), while in more physiological 3D environment they move on helical tracks (11) (Fig. 1C). The ookinetes of Plasmodium and invasive stages of other apicomplexan parasites, such as Toxoplasma tachyzoites and sporozoites, perform similar gliding motility and also tend to have helical trajectories (12, 13). Although the helical trajectories of microgliders resemble the ones predicted for microswimmers (14), the underlying physical mechanisms are fundamentally different, namely substrate- and not fluid-based. Recently, the self-organized surface flow has been investigated for experimentally measured shapes of T. gondii tachyzoites (15) and their motility patterns in structured environments have been characterized by different biophysical methods (16), but the exact relation between cell shape, surface flows, and motility patterns has not been addressed yet.

Fig. 1.

(A) The gliding motility apparatus for Plasmodium sporozoites, the stage of the malaria parasite that infects the vertebrate host. The gliding machinery as sketched here is similar for many organisms from the phylum apicomplexa. The underlying image is a scanning electron micrograph of a sporozoite. While the blue arrows indicate the surface flow of the adhesins, the black arrow indicates the resulting overall cell movement. (B) Circular trajectories of Plasmodium sporozoites moving in 2D on a glass slide, with time coded in color. (C) Helical trajectories in a 3D hydrogel, again with time coded in color. [Scale bars in (B and C) 20 $\mathrm{\mu }$m.] Data courtesy by Mirko Singer and Friedrich Frischknecht.

Gliding motility is not only prominent for apicomplexa, but also for many bacteria (17, 18). Gliding motility of bacteria is based on a larger range of mechanisms than for the apicomplexa, but here, we focus on the ones similar to apicomplexa in that their gliding is based on a surface flow adhesively coupled to the substrate, like the adventurous mode of Myxococcus xanthus or the gliding motility of Flavobacterium johnsoniae (19–29). Both bacteria have the shape of cylindrical rods of 5 to 10 $\mathrm{\mu }$m length and 1 $\mathrm{\mu }$m diameter, and rotate around their long axis while gliding along their body axis, with F. johnsoniae being roughly 50 times faster at 2 $\mathrm{\mu }$m/s. While the machineries generating gliding motility in these bacteria are not yet fully understood, it is known that M. xanthus elastically couples to the substrate (24). Moreover its motor units are rotary and not stationary (30) and move MreB-filaments along helical tracks inside the cells (31). In F. johnsoniae, the motor units are also rotary, but stationary (22). While the molecular basis of gliding motility in bacteria is diverse and not fully understood, the general concept seems to be similar to the case of apicomplexa, namely that spatially distributed force generators effectively move adhesins over the cell surface. Our treatment does not cover twitching motility of bacteria like the social mode of M. xanthus or motility of Neisseria gonorrhoeae, which are based on pilus retraction rather than surface flow and therefore is also more jerky than the gliding motility discussed here.

Given the importance of surface flow for gliding, it is clear that cell shape plays a central role in determining the resulting motility patterns. However, a unifying theory relating cell shape, surface flow, and motility patterns is missing. In the following, we introduce such a theory to describe how the shape of a microglider determines its motility. Importantly, our approach is geometrical in nature and does not depend on the details of the gliding motility apparatus. Motivated by the case of the apicomplexa, we start with the assumptions that gliding motility is driven by independent motor units, which are uniformly distributed below the glider’s surface, and that they self-organize to generate motion through adhesive coupling to a solid substrate. From these assumptions we derive the complete phase behavior of possible motility patterns of microgliders. We then extend the model to also address gliding bacteria, which in contrast to the apicomplexa predominantly move on surfaces and use internal tracks to control surface flow. Again our theory can be used to predict cell trajectories from microscopic rules. In general, it reveals that the surface flows powering gliding always have a strong tendency to rotate the glider in place, and that additional elements are required to avoid pure rotation without productive translocation. We show that curved shape for apicomplexa and prepatterned flow for bacteria would both serve such a function and lead to circular and helical trajectories, exactly as observed for gliding apicomplexa and bacteria. In the future, our theory could also be used to design synthetic microgliders.

Materials and Methods

Geometrical Theory of Gliding.

While the majority of experimental work investigating gliding motility is performed with 2D substrates, the more physiological environment for apicomplexan parasites is 3D. Motivated mainly by this case, we first formulate our theory for 3D environments and later specify it for 2D. We consider gliding motility that is based on adhesion distributed over the complete surface and thus take a continuum approach. We make the following assumptions, as visualized in Fig. 2:

Fig. 2.

Schematic representation of a microglider and its mathematical description. Gliding motility is generated by independent motor units, distributed below the surface of the fixed-shape microglider. The geometry of the glider is described by its surface shape ${\overrightarrow{r}}_{(\theta ,l)}$ and its global movement with translational and rotational velocities $\overrightarrow{V}$ and $\overrightarrow{\mathrm{\Omega }}$, respectively. The motors generate a surface flow of adhesins, ${\mathbf{u}}_{(\theta ,l)}$, which generates friction through the difference to the local relative velocity of the environment, ${\overrightarrow{v}}_{(\theta ,l)}$. This friction determines the global motion and also reorganizes the surface flow.

1. The glider is a rigid body of fixed geometry, which leads to its surface parameterization ${\overrightarrow{r}}_{(\theta ,l)}$, and it performs global motion with translational and angular velocities $\overrightarrow{V}$ and $\overrightarrow{\mathrm{\Omega }}$, respectively. Thus we do not consider any deformations of the cell body.

2. Gliding motility is caused by the surface flow ${\mathbf{u}}_{(\theta ,l)}$ of adhesins, actively driven by a distributed motor machinery.

3. The driving motor system has a fixed target flow speed, which by choice of units is $1$. Because experimental evidence does not suggest otherwise, we assume that the driving is isotropic, that is individual motors do not have any preferred direction.

4. The surface motors tend to align their direction with the relative movement direction of the environment. This means that the mismatch between the motor target velocity and the actual environment velocity does not only create forces but also feeds back into the motor configuration, establishing a mechanism for adaptive self-organization.

The existence of the surface flow field ${\mathbf{u}}_{(\theta ,l)}$ is not only the simplest assumption to explain gliding motility, it also has been directly measured by particle tracking (16, 20, 32) or indirectly by fluorescence microscopy of moving parts of the motor machinery (15, 21). Due to the small density of adhesins in the membrane, this flow is assumed to be infinitely compressible (or, equivalently, pressure-free). Moreover, the density of the adhesins is assumed to be constant, resulting in a constant coupling strength between surface flow and environment over the whole surface.

In order to describe the coupling between the cell and environment, we first consider how movement determines surface flow; then we consider how the surface flow determines movement. The surface flow has to be compared with the relative velocity of the environment to a surface element, ${\overrightarrow{v}}_{(\theta ,l)}$, which depends on the total translational and angular velocities through

$${\overrightarrow{v}}_{\left(\theta ,l\right)}=-\left(\overrightarrow{V}+\overrightarrow{\text{Ω}}\times {\overrightarrow{r}}_{\left(\theta ,l\right)}\right).$$ [1]

The relative velocity can be further decomposed into tangential and normal parts by means of the local projection into the surface:

$$\begin{array}{c}{v}_{(\theta ,l)}^{\Vert }=P_{(\theta ,l)}{\overrightarrow{v}}_{(\theta ,l)},{\overrightarrow{v}}^{\perp }=\left(1-P_{(\theta ,l)}^T{P}_{(\theta ,l)}\right)\overrightarrow{v}.\end{array}$$ [2]

Here bold symbols, like ${\mathbf{u}}_{(\theta ,l)}$ and ${\mathbf{v}}_{(\theta ,l)}^{\Vert }$, denote 2-component vector fields in the tangent bundle of the surface, while usual vector arrows mark 3-component vectors. A tangential 2-component vector at coordinates $(\theta ,l)$ can be embedded into the lab-frame by $P_{(\theta ,l)}^T$. Assuming linear feedback of friction from the difference between surface flow ${\mathbf{u}}_{(\theta ,l)}$ and tangential environmental velocity ${\mathbf{v}}_{(\theta ,l)}^{\Vert }$ into the motor machinery with a coupling constant $\mathrm{\Gamma }$, as well as a third-order active driving term with driving strength $\eta$ as known from Vicsek or other flocking models (15, 33, 34), we arrive at the following generic evolution equation for the surface flow:

$$\begin{array}{c}\hfill {\partial }_t{\mathbf{u}}_{(\theta ,l)}=-\mathrm{\Gamma }\left({\mathbf{u}}_{(\theta ,l)}-{\mathbf{v}}_{(\theta ,l)}^{\Vert }\right)+\eta {\mathbf{u}}_{(\theta ,l)}\left(1-|{\mathbf{u}}_{(\theta ,l)}{|}^2\right).\end{array}$$ [3]

Through ${\mathbf{v}}^{\Vert }$, this equation depends on the global motion, described by $\overrightarrow{V}$ and $\overrightarrow{\mathrm{\Omega }}$. There are numerous additional contributions that could be included, as advection of the polarized component of the motor machinery (and hence the velocity), or local ordering. While relevant to the dynamics in the absence of coupling to the environment (15), we omit them here in our analytical theory. If the coupling with the environment is strong, as assumed in the following, these additional local contributions will be negligible compared to the global geometric constraints. In the numerical simulations, however, we can introduce these terms, and verify that their influence on the stationary solutions is small, as shown in SI Appendix, cf. SI Appendix, Fig. S3.

We next consider how the surface flow leads to movement, thus closing the equations of our gliding model. The mismatch between surface flow and environment velocity in the tangent space creates frictional forces with a friction constant $\gamma$, leading to a driving force resulting from integration over the surface:

$$\begin{array}{c}{\overrightarrow{F}}^{\left|\right|}=-\gamma {\displaystyle \int dA{P}_{(\theta ,l)}^T\left[u_{(\theta ,l)}-v_{(\theta ,l)}^{\Vert }\right]}.\end{array}$$ [4]

The environmental velocity normal to the surface also creates friction, for which we introduce a dimensionless factor $\zeta$ to allow for different friction behavior between tangential and normal velocity:

$$\begin{array}{c}{\overrightarrow{F}}^{\perp }=\zeta \gamma \int dA{\overrightarrow{v}}_{(\theta ,l)}^{\perp }.\end{array}$$ [5]

In the context of adhesive friction in a complex solid environment, it is an unknown parameter that we assume to be of order unity. An estimate for the related friction anisotropy in M. xanthus is also in this range (35). For microswimmers, the global analogon $\zeta$ has the typical value of $2$ (1). Assuming overdamped dynamics, for a given surface flow $\mathbf{u}$ the rigid body dynamics will immediately adapt and $\overrightarrow{V}$ and $\overrightarrow{\mathrm{\Omega }}$ are determined according to the force balance

$$\begin{array}{c}{\overrightarrow{F}}^{\left|\right|}(u_{(\theta ,l)},\overrightarrow{V},\overrightarrow{\Omega })+{\overrightarrow{F}}^{\perp }(u_{(\theta ,l)},\overrightarrow{V},\overrightarrow{\text{Ω}})=0.\end{array}$$ [6]

In this overdamped description, $\overrightarrow{V}$ and $\overrightarrow{\mathrm{\Omega }}$ are not dynamical quantities, but are instantaneously determined by the surface flow field ${\mathbf{u}}_{(\theta ,l,t)}$, which is the only dynamical field of the theory. The global motion is determined by an integral of the surface flow, making this an integro-differential equation. For the torques, we have a similar balance equation as for the forces

$$\begin{array}{c}{\overrightarrow{M}}^{\left|\right|}(u_{(\theta ,l)},\overrightarrow{V},\overrightarrow{\text{Ω}})+{\overrightarrow{M}}^{\perp }(u_{(\theta ,l)},\overrightarrow{V},\overrightarrow{\text{Ω}})=0,\end{array}$$ [7]

thus completing our model definition.

Stationary Solutions.

We start our discussion of the solutions of our gliding model with the stationary solutions of the dynamical equation Eq. 3. We immediately see that these solutions require a parallel orientation of the surface flow to the environmental velocity, that is ${\mathbf{u}}_{(\theta ,l)}||{\mathbf{v}}_{(\theta ,l)}^{\Vert }$ everywhere. This reduces it to a scalar equation, which as a third-order polynomial can be solved analytically. We now make the assumption of strong coupling to the environment, $g=\mathrm{\Gamma }/\eta \gg 1$, which means that an adhesin, bound on one side to the environment and on the other side subject to the motor machinery of the glider, will mostly be stationary with respect to the environment, as indeed observed for clusters of adhesins in apicomplexan parasites as well as in gliding bacteria (9, 19, 21). Under this assumption, the only physically relevant, positive, real solution is

$$\begin{array}{c}\hfill \mathbf{u}({\mathbf{v}}^{\Vert })={\mathbf{v}}^{\Vert }+\frac{1-|{\mathbf{v}}^{\Vert }{|}^2}{g}{\mathbf{v}}^{\Vert }.\end{array}$$ [8]

We conclude that the surface flow generally follows the environment, but will be slightly faster and driving the system if the tangential environmental speed ${\mathbf{v}}^{\Vert }$ is smaller than the motors’ target speed $1$. Otherwise, it will be slower and reduce the motion. Overall, this defines a stable stationary solution of the system.

With Eq. 8, we can now eliminate the surface flow $\mathbf{u}$ from the force balance, Eq. 6. We introduce the generalized 6-component velocity $\overrightarrow{W}=(\overrightarrow{V},\overrightarrow{\text{Ω}})$ and find

$$\begin{array}{cc}{\overrightarrow{F}}^{\left|\right|}& =-\gamma \int dA{P}^{T}P\left[\frac{1-|v^{\Vert }{|}^2}{g}\overrightarrow{v}\right]\\ & =\frac{\gamma }{g}\int dA\left(1-{\overrightarrow{W}}^T{G}_{(\theta ,l)}\overrightarrow{W}\right)\\ & \times P^{T}P\left[\overrightarrow{V}+\overrightarrow{\text{Ω}}\times {\overrightarrow{r}}_{(\theta ,l)}\right].\end{array}$$ [9]

Here the 6$\times$6 geometrical kernel $G_{(\theta ,l)}$ arising from $|{\mathbf{v}}^{\Vert }{|}^2$ is defined as

$$G_{(\theta ,l)}=\left(\begin{array}{cc}{P}^{T}P& P^{T}P{\left[\overrightarrow{r}\right]}_{\times }^T\\ {\left[\overrightarrow{r}\right]}_{\times }{P}^{T}P& {\left[\overrightarrow{r}\right]}_{\times }{P}^{T}P{\left[\overrightarrow{r}\right]}_{\times }^T\end{array}\right)$$ [10]

and combines the surface orientation information $P_{(\theta ,l)}$ relevant to determine the tangential part with the position information ${\overrightarrow{r}}_{(\theta ,l)}$ in form of the matrix representation of the cross-product ${[\overrightarrow{r}]}_{\times }\overrightarrow{a}=\overrightarrow{r}\times \overrightarrow{a}$, necessary to compute the effect of rotation (details in SI Appendix).

The geometric part of the problem can now be condensed into the geometry tensors

$$\begin{array}{cc}\hfill {\mathcal{G}}^{\mathit{ij}}=& \int dA{G}_{(\theta ,l)}^{\mathit{ij}},\hfill \end{array}$$ [11]

$$\begin{array}{cc}\hfill {\mathcal{G}}_2^{\mathit{klij}}=& \int dA{G}_{(\theta ,l)}^{\mathit{kl}}{G}_{(\theta ,l)}^{\mathit{ij}}.\hfill \end{array}$$ [12]

The substructure of three velocity and three angular velocity components of $\overrightarrow{W}$ gives $\mathcal{G}$ a block matrix structure. The upper left submatrix can be understood as the mean projection of a vector given in the lab frame onto the surface, while the other block matrices are weighted with the distance vector to the origin. As one can see from Eq. 10, $\mathcal{G}$ is symmetric. Its transformation behavior and relation to symmetries of the underlying geometry are similar to the ones for the hydrodynamic resistance tensor (36) (details in SI Appendix). Identifying the remaining geometry-dependent part of Eq. 9 as the top row of $G_{(\theta ,l)}\overrightarrow{W}$, and the bottom row as the similar term that would arise in the respective expression for ${\overrightarrow{M}}^{\left|\right|}$, we find for the forces and torques:

$$\begin{array}{cc}{\left(\begin{array}{c}{\overrightarrow{F}}^{\left|\right|}\\ {\overrightarrow{M}}^{\left|\right|}\end{array}\right)}_i& =\frac{\gamma }{g}\int dA\left(1-{\overrightarrow{W}}^T{G}_{(\theta ,l)}\overrightarrow{W}\right)G_{(\theta ,l)}^{ij}{W}_j\\ & =\frac{\gamma }{g}\left({\mathcal{G}}^{ij}-{\mathcal{G}}_2^{ijkl}{W}_k{W}_l\right)W_j.\end{array}$$ [13]

Repeating the same line of argument for the normal component (Eq. 5), a third geometry tensor ${\mathcal{R}}^{\mathit{ij}}$ emerges, similar to ${\mathcal{G}}^{\mathit{ij}}$, but replacing the projections $P$ in Eq. 10 with identity operators, such that the normal space can be expressed by the difference of $\mathcal{R}$ and $\mathcal{G}$:

$$\begin{array}{c}{\left(\begin{array}{c}{\overrightarrow{F}}^{\perp }\\ {\overrightarrow{M}}^{\perp }\end{array}\right)}_i=-\zeta \gamma \left({\mathcal{R}}^{ij}-{\mathcal{G}}^{ij}\right)W_j.\end{array}$$ [14]

Finally, we can rewrite the force and momentum balance Eqs. 6 and 7 as a condition on the global motion $W$,

$$\begin{array}{c}\hfill \left({\mathcal{G}}^{\mathit{ij}}-{\mathcal{G}}_2^{\mathit{klij}}{W}_k{W}_l\right)W_j-\zeta g\left({\mathcal{R}}^{\mathit{ij}}-{\mathcal{G}}^{\mathit{ij}}\right)W_j=0,\end{array}$$ [15]

which effectively is a third-order polynomial in $\overrightarrow{V}$ and $\overrightarrow{\mathrm{\Omega }}$. Its coefficients encode the geometric shape of the glider and its solutions determine the full stationary field configuration of the surface flow via Eqs. 2 and 8. Furthermore, these solutions depend only on one effective parameter, $c=\zeta g=\zeta \mathrm{\Gamma }/\eta$. This $c$ is small for strong driving, weak coupling, and weaker normal than tangential friction, and large for weak driving, strong coupling, and stronger normal than tangential friction. Our theory as discussed here is an approximation for the large coupling case. The geometry tensors can be calculated analytically for many shapes of interest (for the simplest case of an open cylinder, the results are documented in SI Appendix).

Stability Analysis.

A linear stability analysis in regard to the surface flow ${\mathbf{u}}_{(\theta ,l)}$ based on Eq. 3 shows that the stationary solutions are stable as long as the global motion $\overrightarrow{W}$ is not affected. This makes sense because we already have seen from Eq. 8 that mechanisms exist to stabilize the surface flow. Therefore it is a better question to ask about global stability: If by an external influence (e.g., a collision with an obstacle), the surface flow is globally changed with a resulting deviation in $\overrightarrow{W}$, will the surface flow afterward relax back toward the previous configuration, necessarily also reverting $\overrightarrow{W}$ to its previous value, or not? To answer this question, we start with a perturbation $\delta \overrightarrow{W}$ around a solution $\overrightarrow{W}$ with corresponding surface flow ${\mathbf{u}}_{(\theta ,l)}$ and look for the change

$$\begin{array}{c}\hfill \delta {\mathbf{u}}_{(\theta ,l,t)}=-P_{(\theta ,l)}\left(\delta \overrightarrow{V}+\delta \overrightarrow{\mathrm{\Omega }}\times {\overrightarrow{r}}_{(\theta ,l)}\right).\end{array}$$ [16]

As a measure for the effect of the perturbation, we will investigate the dynamics of

$$\begin{array}{c}\hfill \mathrm{\Delta }(t)=\int dA{\left|\delta {\mathbf{u}}_{(\theta ,l,t)}\right|}^2,\end{array}$$ [17]

which upon expanding in small perturbations and large coupling $c$ is found to be

$$\begin{array}{cc}\hfill {\partial }_t\mathrm{\Delta }(t)& \approx 2{\mathcal{J}}^{\mathit{ij}}\delta W_i\delta W_j\hfill \end{array}$$ [18]

$$\begin{array}{cc}\hfill {\mathcal{J}}^{\mathit{ij}}(\overrightarrow{W})& =[g{\left(\mathcal{G}{\left(\zeta \mathcal{R}+(1-\zeta )\mathcal{G}\right)}^{-1}\mathcal{G}\right)}^{\mathit{ij}}\hfill \\ \hfill & +(1-g){\mathcal{G}}^{\mathit{ij}}-{\mathcal{G}}_2^{\mathit{ijkl}}{W}_k{W}_l-2{\mathcal{G}}_2^{\mathit{ikjl}}{W}_k{W}_l].\hfill \end{array}$$ [19]

A stationary solution of global movement $\overrightarrow{W}$ is hence predicted to be stable if the matrix $\mathcal{J}(\overrightarrow{W})$ is negative definite. As this procedure probes only a subspace of possible perturbations and only uses the total perturbation size $\mathrm{\Delta }(t)$ to decide about growth or decay of a particular perturbation, our global stability analysis is not complete. We therefore complement it by numerical simulations that directly solve Eqs. 3–7 with appropriate discretizations (details in SI Appendix). In general, we find excellent agreement with the predictions of our stability analysis. In contrast to the analysis of the stationary solutions, we find that the stability depends directly on the individual values of $g$ and $\zeta$, and not only on their product (in the following, we use $\zeta =1/2$). This results from the fact that while the stationary problem could be reduced to the finite space of the rigid-body motion described by Eq. 15, the underlying dynamical problem is still that of the infinite dimensional surface flow field ${\mathbf{u}}_{(\theta ,l)}$.

Results

Overview.

In the following, we apply our geometrical theory of gliding motility to a range of different cell shapes, starting with rods and ellipsoids and continuing with curved rods. Afterward, we investigate the motion on a 2D substrate, with particular attention to gliding bacteria. In general, all our results are analytical solutions to the central equation Eq. 15. For each case of interest, the geometry tensors ${\mathcal{G}}^{\mathit{ij}}$, ${\mathcal{G}}_2^{\mathit{ijkl}}$, and ${\mathcal{R}}^{\mathit{ij}}$ are calculated analytically with the computer algebra software Mathematica. Once a solution is obtained in terms of the global movement $\overrightarrow{W}$, we can find the corresponding surface flow field ${\mathbf{u}}_{(\theta ,l)}$. Stability of the solutions is decided according to the matrix $\mathcal{J}(\overrightarrow{W})$ from Eq. 19. Very importantly, all our analytical results are verified by numerical solutions.

Spherocylindrical Cell Shape.

The spherocylinder is the open-ended cylinder closed by two spherical caps and can be readily treated by our theory. Due to the three symmetry planes, the geometry tensor $\mathcal{G}$ has to be diagonal. With the radius set to $1$, aspect ratio $\alpha$, and the axis of symmetry in $z$-direction, it can be written as

$$\begin{array}{c}\hfill \text{Diag}(\mathcal{G})=\frac{2\pi }{3}\left(\begin{array}{c}3\alpha +1\\ 3\alpha +1\\ 6\alpha -2\\ {\alpha }^3+{\alpha }^2+4\alpha -2\\ {\alpha }^3+{\alpha }^2+4\alpha -2\\ 6\alpha -2\end{array}\right).\end{array}$$ [20]

We find only two solutions, namely pure rotation (only ${\mathrm{\Omega }}_z$ nonzero) and pure translation (only $V_z$ nonzero):

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_z^{\text{rot.}}=\sqrt{\frac{15\alpha -5}{15\alpha -7}},& V_z^{\text{trans.}}=\sqrt{5}\sqrt{\frac{3\alpha -c-1}{15\alpha -7}}.\hfill \end{array}$$ [21]

The respective surface flows generating these motions are displayed in Fig. 3A. For large aspect ratio $\alpha$, both values approach unity, because for very elongated shapes, the driving force created at the tangential part of the surface dominates over the drag created at the ends (note that both velocity and radius are normalized to $1$, thus also normalizing angular velocity). For aspect ratio approaching $1$ (the case of a sphere), the angular velocity actually becomes larger than $1$, because now a larger portion of the surface drives the rotation from positions close to the axis of rotation.

Fig. 3.

Steady state solutions for rotationally symmetric shapes, namely a spherocylinder and an ellipsoid, in a 3D environment. (A) The surface flows for the three types of solutions identified by the theory: pure rotation (blue), pure translation (red), and mixed motion (green) (Movies S1 and S2). (B) Bifurcation diagram for the spherocylinder and (C) bifurcation diagram for the ellipsoid. The Top row shows velocity in direction of the long axis and the Bottom row angular velocity around that axis. The three fundamental solutions are distinguished by the same colors used in (A). Dashed and solid lines are unstable and stable solutions, respectively. Both the aspect ratio $\alpha$ and the coupling $c$ of the surface flow to the environment are varied as bifurcation parameters. The bifurcation diagrams are symmetric under sign change and only the positive branches are shown. Black circles denote pitchfork bifurcations from the trivial solution, which is not shown. Orange circles denote pitchfork bifurcations. Diamonds are steady states found by numerical simulation and are in excellent agreement with the analytical results.

We next investigated the stability of the two stationary solutions and found that pure rotation is always stable, while pure translation is always unstable. The corresponding bifurcation diagrams are shown in Fig. 3B, with solid and dashed lines for stable and unstable solutions, respectively (compare also Movie S1). The translational solution bifurcates from the trivial solution with increasing aspect ratio, i.e., it does not exist for a near-spherical shape, where the surface area parallel to the motion used for driving is comparable to the perpendicular area, generating friction. The translating solution also bifurcates into the trivial solution for higher coupling, i.e., weaker driving. The trivial solution (vanishing surface flow and hence no motion) remains unstable before and after both these bifurcation points, as it is still unstable to perturbations in the direction of the rotational solution. Other solutions, such as rotations around another axis, arise only at $c<1$, outside the range in which our theory is valid, or at aspect ratios $\alpha \approx 1$, where the solutions become degenerate as the geometry approaches a sphere.

The fact that the spherocylindrical glider in 3D prefers the rotating solution can be understood from an energy perspective: For the rotating cylinder, the surface is fixed in space, as the surface flow compensates the underlying rotation, hence minimizing drag and dissipation. For a biological system, which invests energy to drive the surface flow in order to move, the global stability of the solution with pure rotation is a fundamental problem. In the following, we will investigate which different shapes can help to mitigate the tendency of surface flow motors to rotate the glider in place.

Ellipsoidal Cell Shape.

The ellipsoid, more specifically a prolate spheroid with short semiaxis $1$ and long semiaxis $\alpha$, shows similar rotational and translational solutions as the spherocylindrical glider. Strikingly, however, now a third mode becomes possible, namely a mixed solution with a tilted flow field, as shown in Fig. 3A.

In Fig. 3C, we show the corresponding bifurcation diagrams. The rotational and translational solutions in Fig. 3C look similar to the previous case of the spherocylinder. The translational solution only deviates by the bifurcation into the trivial solution being shifted to even higher coupling, caused by the smaller proportion of perpendicular surface area for the ellipsoid compared to the spherocylinder. The rotating solution bifurcates into the new mixed solution for increasing aspect ratio or decreasing coupling, made possible by the varying radius of the ellipsoid: While for the spherocylindrical glider, the radius is constant and the surface flow can be at its target speed everywhere except at the caps, for the ellipsoidal glider the varying distance from the axis of rotation creates a mismatch. To compensate, additional flow in $z$-direction is created where the radius and hence the velocity due to rotation is smaller, causing translation. As all bifurcations in our theory are symmetric under velocity reversal, this solution spontaneously breaks symmetry to decide in which direction the glider starts to move.

Curved Cell Shape.

The most obvious solution to overcome the pure rotation solution is to break rotational symmetry of the shape, which is achieved, e.g., by a curved rod. This suggests immediately that Plasmodium sporozoites have evolved curvature in order to avoid the stability of the rotational solution.

We define the geometry of the curved rod as a torus segment of curvature angle $\mathrm{\Phi }$, closed by spherical caps like the straight spherocylinder. The surface flow for this shape allows in principle the same three solutions as previously discussed and now shown in Fig. 4A. However, due to the curved shape, the rotational solution becomes coupled to a $V_y$ contribution, offsetting the axis of rotation, while the translational solution becomes coupled to an ${\mathrm{\Omega }}_y$ rotation, leading to circular trajectories (Movie S3). The mixed motion here has nonzero $V_y,V_z,{\mathrm{\Omega }}_y$, and ${\mathrm{\Omega }}_z$, and because the coefficients are very complex combinations in the free parameters, the aspect ratio $\alpha$, curvature angle $\mathrm{\Phi }$ and coupling $c$, the solutions for the mixed state here were obtained as numerical roots of the polynomial Eq. 15.

Fig. 4.

Steady state solutions for a curved spherocylinder (segment of a torus with spherical caps), which breaks rotational symmetry, gliding in a 3D environment. (A) The surface flows for the three types of solutions identified by the theory (see Movies S3 and S4 for example trajectories). (B) Bifurcation diagrams. The Top row shows velocity in direction of the long axis and Bottom row angular velocity around that axis. Colors again as in (A). The trivial solution is plotted in gray, but only in the Middle column where it is displaced below 0 for better visibility where necessary, otherwise it remains unstable and is omitted. Dashed and solid lines are unstable and stable solutions, respectively. The bifurcation parameters are the aspect ratio $\alpha$, the coupling $c$ of the surface flow to the environment, and the curvature angle $\mathrm{\Phi }$ as described in the Inset. The bifurcation diagrams are symmetric under sign change, only the positive branches are shown. Black circles mark pitchfork bifurcations from the trivial solution, and orange circles pitchfork bifurcations between the displayed solutions. Diamonds are steady states found by numerical simulation and are in excellent agreement with the analytical results.

To keep the bifurcation diagrams simpler, we only present the $z$-quantities in Fig. 4B. The dependent $y$-quantities are shown in SI Appendix, Fig. S2. In general, we find that the mixed solution bifurcates from the rotational solution as before but now leads into a stable translational solution with increasing aspect ratio or curvature, or decreasing coupling. Taking a closer look at the coupling in Fig. 4B, we notice that high coupling leads to another bifurcation of the rotational solution into the trivial solution. Opposed to the rotationally symmetric shapes, here the trivial state becomes stable at sufficiently high coupling, as the asymmetry prevents friction-free rotation.

To represent the complete solution space of our theory, in Fig. 5, we have assembled phase diagrams. In Fig. 5A, we show as a reference the phase diagram for the previously discussed ellipsoid, which has stability regions for both rotating (blue) and mixed (green) solutions. From the analytical solutions, the critical value of coupling $c_{\text{Crit.}}$ separating rotational and mixed solution can be approximated as

$$\begin{array}{c}\hfill c_{\text{Crit.}}(\alpha )\approx \frac{8(\alpha -1)}{35}+\frac{596{(\alpha -1)}^2}{3675}+\mathcal{O}\left({(\alpha -1)}^3\right),\end{array}$$ [22]

Fig. 5.

(A) Phase diagram for the ellipsoid as a function of aspect ratio $\alpha$ and coupling $c$ ($\zeta =1/2$), showing the transition from stable rotating solution (blue) to stable mixed solution (green), with the analytical critical value Eq. 22 (orange). Compare SI Appendix, Fig. S1 for different values of $\zeta$. (B–D) Cross-sections of the parameter space of stable solutions for the curved spherocylinder, at fixed curvature angle $\mathrm{\Phi }$ (B), coupling $c$ (C) or aspect ratio $\alpha$ (D). Dash-dotted, dotted, and dashed lines correspond to the planes intersecting each other and the conditions shown in the first, second, and third column of Fig. 4, respectively. (E) Helical trajectory of the mixed state at $\mathrm{\Phi }=1/3$, $c=3$, and $\alpha =10$, in blue the trajectory of the center, in red that of an off-axis point at the rear of the shape. See also Movie S4. (F) Maximum intensity projection in $z$ and time for fluorescence microscopy images of a Plasmodium sporozoite moving through a 3D hydrogel, exhibiting a helical trajectory. Data courtesy of Mirko Singer and Friedrich Frischknecht.

which is shown in orange in Fig. 5A. From the phase diagrams for the curved spherocylinder, Fig. 5B–D, we see that now the solution with translation (red regions), which leads to circular trajectories, is very prominent. As before, there are also parameters for which only rotation can occur (blue regions). The mixed state (green region), which leads to helical trajectories, usually occurs as a transitional state between rotation and translation, with the transition taking place at higher curvatures for lower aspect ratios (Fig. 5C). This can be understood as the influence of curvature being more pronounced in a more slender shape. At high coupling and fixed curvature, Fig. 5B additionally reveals that the transition in aspect ratio can also have an intermediate regime where the trivial state is stable, i.e., a thick rod rotates, an intermediate thickness is motionless, and a thin rod translates. For fixed aspect ratio, the coupling determines whether with increasing curvature the rotational solution transitions into the translational (low coupling) or trivial solution (high coupling). At the bifurcation points and boundaries more complicated solutions arise, including, e.g., a rotation around an additional axis, which we omitted above as the stable solutions we discuss cover the vast majority of the phase space. In Fig. 5E (compare Movie S4) we show one of the helical trajectories occurring in the mixed state (the blue and red lines track the center and an off-axis point on the surface, respectively). Indeed such helical trajectories are very common for Plasmodium sporozoites in 3D environments (Fig. 1C) and a close-up for an experimental example is shown in Fig. 5F.

2D Gliding of Apicomplexa.

Many experiments on apicomplexa are performed on 2D substrates rather than in their natural 3D environments. Therefore we now consider the 2D case as a limit of our 3D theory (Fig. 6A). To this end, we have to make two changes: First, the geometry tensors ${\mathcal{G}}^{\mathit{ij}}$, ${\mathcal{G}}_2^{\mathit{ijkl}}$, and ${\mathcal{R}}^{\mathit{ij}}$, previously obtained by integration over the whole surface, are now computed only for the part of the glider’s surface that is sufficiently close to the substrate to interact adhesively, described by a contact angle ${\delta }_S$ (cf. Fig. 6B). Second, assuming the body keeps maximal contact with the substrate, the degrees of freedom of the rigid body movement are restricted.

Fig. 6.

Gliding of a curved rod on a substrate at aspect ratio $\alpha$ = 10 and curvature angle $\mathrm{\Phi }$ = 1. (A) Schematics and definition of the trajectory radius $R_T$ and radius of curvature $R_C$. (B) Definition of the contact angle ${\delta }_S$. (C and D) Solutions and radius mismatch as function of contact angle ${\delta }_S$ (C) and coupling $c$ (D).

The curved spherocylinder on a substrate (Fig. 6A) resembles the well-established gliding assay as displayed for Plasmodium sporozoites in Fig. 1B. We assume the body is oriented and stays oriented maximizing contact area, hence allowing only rotation perpendicular to the substrate. We find stable translation on shape-dependent circular trajectories and analyze these by comparing the trajectory radius $R_T$ to the glider’s radius of curvature $R_C$. We find that generally $R_T$ is larger than $R_C$. This effect is partly due to the friction at the caps generating torque against turning, hence increases with larger contact angle ${\delta }_S$ (Fig. 6C). Second, the flow on the inner side of the contact area has a slightly smaller effective radius, but the same target speed, resulting in an increasing trajectory radius for low coupling (Fig. 6D). The analytical prediction is once again confirmed by the simulation, showing that our theory can include subtle effects as the distribution of the force generation over different radii (i.e., inside vs. outside of contact area) and the torque generated by friction at the spherical caps, all as a function of contact angle ${\delta }_S$.

2D Gliding of Bacteria.

Finally, we investigate the straight spherocylinder on a substrate, a situation resembling the geometry of gliding bacteria, which often move on 2D surfaces. F. johnsoniae and M. xanthus are both known to rotate while gliding on a substrate (21, 25, 30). A direct application of our theory finds again that the only stable solution is rotation in place (SI Appendix, Fig. S4), without productive gliding. Because these microgliders are not curved and axisymmetric, other mechanisms must be at play to avoid the stability of the rotating solution.

From the assumptions made for the general 3D theory motivated by the apicomplexa, the most obvious aspect that does not fit to bacteria is the assumption that the motor-driven movement of the adhesins is isotropic. In fact, it is well known that the propulsion of the adhesins in bacteria is more directed, often along internal tracks of a helical geometry as shown in Fig. 7A. We therefore start by introducing a small directional bias $B$ to the motors, which is dimensionless and measured relative to the driving strength $\eta$, with a detailed derivation provided in SI Appendix. We can analytically solve the resulting dynamic equation for its stable states, depending on aspect ratio $\alpha$, coupling strength $c$, contact angle ${\delta }_S$, and bias strength $B$. For suitable parameters, a stable mixed solution as shown in Fig. 7B (compare SI Appendix, Fig. S4 and Movie S5) emerges. This demonstrates that anisotropic flow stabilizes forward motion. Furthermore, the flow field of the mixed state produces flow lines (green in Fig. 7B) that follow the observed helical track in bacterial surface flows in one direction.

Fig. 7.

Bacterial gliding on surfaces. (A) Schematic of the gliding of bacteria, gliding forward while simultaneously rotating. Experiments suggest Flavobacterium johnsoniae and Myxococcus xanthus organize the flow along opposing helical tracks, possibly with only one track coupled for productive motility. (B) With a small bias along the long axis, we obtain self-organized surface flow patterns resulting in simultaneous rotation and translation. This generates the one-way helical flow lines along the surface (green) shown in (A). The Inset shows a cross-section and the definition of the contact angle ${\delta }_S$. (C) Bifurcation diagram displaying the solutions for the translational velocity along (red) and rotational velocity around (blue) the long axis of the cylinder, as function of bias $B$ with fixed ${\delta }_S=\pi /4$ and $\alpha =c=10$. Orange circles mark pitchfork bifurcations, and diamond symbols results of numerical simulations. (D) Same as (C), but now bifurcation diagram as function of contact angle ${\delta }_S$ at fixed bias $B=0.01$. (E) Translational velocity in the coupling $c$/bias $B$ phase space. (F) Prepatterned surface flow with helical pattern turning around the cell body in opposite directions as illustrated in (A), with colormap showing coupling strength (Movie S7). (G) Resulting mean translational (red) and angular (blue) velocities for prescribed helical surface flows as shown in (F).

Investigating the parameter dependence in more detail, the bifurcation diagram Fig. 7C demonstrates that effective motility can be obtained with small bias values of around 0.01, suggesting a small anisotropy is sufficient to shift the stable solution to mixed translation and rotation. At a finite bias, a pitchfork bifurcation occurs and the rotation vanishes completely. As discussed in SI Appendix, the stability theory introduced before remains valid for small $B$, but we can observe that the stability change of the pure translation branch after the pitchfork bifurcation at higher bias is not captured anymore, and corrections for the bias in the stability theory are necessary. The bifurcation as function of contact angle ${\delta }_S$ in Fig. 7D is largely the inverse; at large contact angle, the higher friction at the caps favors rotation. This suggests a more rigid shape, which reduces contact area, makes it easier to glide productively. The relation between coupling $c$ and bias $B$ in Fig. 7E shows that the higher the coupling the stronger the anisotropy has to be in order to obtain the same translational velocity. This is similar to a recently found trade-off in twitching motility of Pseudomonas aeruginosa, which need to balance their adhesion in order to avoid being ripped off the surface, but still be able to migrate (37).

The helical surface flow lines used by this model are not closed, they only move from the front to the back. We can also consider a prepatterning in the model, replacing the self-organization with a detailed patterning of the surface flow consisting of intertwined bidirectional helical trajectories (Fig. 7F). The opposing tracks would generate forces canceling each other. We additionally assume that only one of the two directions is coupled to the substrate (Fig. 7A), a mechanism investigated in a simplified 1D model in ref. 38. Varying the number of helical turns along the cell length, the model predicts the resulting motility pattern (Fig. 7G and Movie S7). The more helical turns are included, the more dominant rotational motion becomes compared to translation. The absolute prepatterning neglects the dynamic component of the surface flow but demonstrates that these surface flows, once established, are mechanistically consistent with the observed global motion.

Discussion

Here we have presented a universal and geometrical theory of how active surface flow leads to the motility patterns of microgliders. The overall thrust of this theory is similar to the question how the local movements of microswimmers lead to their large-scale motility patterns. We found that the central concept required to answer this question is the definition of geometrical quantities (the three geometry tensors ${\mathcal{G}}^{\mathit{ij}}$, ${\mathcal{G}}_2^{\mathit{ijkl}}$, and ${\mathcal{R}}^{\mathit{ij}}$) that have similar transformation properties like the resistance tensors for microswimmers. Otherwise, however, the underlying physics is very different, and therefore, we also find unique answers that have not been described before.

As we show here, the three geometry tensors can be calculated analytically for all shapes of interest (spherocylinders modeling rod-like bacteria, ellipsoids modeling more bulky shapes like Plasmodium ookinetes and curved spherocylinders modeling Plasmodium sporozoites and Toxoplasma tachyzoites). Inserting them into the force balance, Eq. 15, we succeeded in analytically deriving stationary solutions and their stability, in excellent agreement with the results from numerical simulations.

Our theory reveals a tight interaction between rotation and translation for gliding microorganisms. In a simple axisymmetric shape, the rotational solution is dominant and has to be suppressed for productive forward motion. This might be achieved by curved shapes, like for Plasmodium sporozoites. The curved shape of sporozoites leads to circular and helical trajectories. Helical trajectories allow for productive motility, but exist only in 3D environments, hence curved shapes might not be as favorable for organisms that require motility on 2D substrates. The helical trajectories of the mixed state of the curved spherocylinders resemble the experimentally observed motility patterns of Plasmodium sporozoites and Toxoplasma tachyzoites (11, 39). Although other reasons exist that might have favored the evolution of such helical trajectories, like circling around blood vessels or efficient search strategies in complicated 3D environments (40, 41), our universal theory makes a strong case for the need to avoid rotation by shape control. A notable exception seems to be the case of certain gregarines, which have been observed to glide on straight paths on 2D substrates (42). However, at several hundred $\mathrm{\mu }$m in length, these parasites are one to two orders of magnitude larger compared to gliding Plasmodium or Toxoplasma gliders, thus several additional effects might come into play, most prominently the effects of cell deformability. Our theory assumes a rigid shape that leads to instantaneous information transfer across the whole cell body; in practice, even Plasmodium sporozoites are known to be rather flexible (43).

Another limitation of our theory is that helical trajectories emerge only for the mixed state of curved spherocylinders, while the phase diagrams from Fig. 5B–D show larger stability regions for rotating and circular solutions. This might be in part due to the assumption of a rigid shape. Allowing the shape to twist and adapt to the helical trajectory could widen the parameter region where the mixed solution is stable. Generally, deformation is another layer of complexity to the motility situation, as many of the organisms described here are observed to deform, at times strongly, e.g., kink, during motility. However, they usually return to their prescribed shape rather quickly. Hence, we believe that our theory is a valid reference case, upon which deformability acts as an additional degree of freedom. The (in)stability behavior of deformability might open interesting insights into interactions with obstacles and path finding in constraint environments but has to be studied by more complex numerical models.

We further note that our theory predicts bifurcations that are invariant under velocity reversal and therefore symmetric in regard to the direction of motion, while many biological systems of interest are polarized and favor one direction of motion (including a preference for a certain chirality). Yet our theory provides an interesting reference against which one now can discuss potential mechanisms to break this symmetry. An obvious one would be to abandon the isotropic activity of the motors and to give them a preferred direction; however, for sporozoites, there is no experimental evidence in this direction. This suggests that the self-organized nature of the surface flow has other advantages; in particular, it might be needed to quickly switch motility patterns if the glider runs into a mechanically difficult situation. Another and possibly more likely way to control the direction of surface flow is the secretion of the adhesins, which is known to be strongly polarized in apicomplexa (secreted at the apical ring at the front and severed by enzymes at the back). This aspect should be addressed in future work, but will be difficult to represent in an analytical theory.

Extending the theory as developed for apicomplexa to bacteria, which typically move on external surfaces, we find that also on 2D substrates the rotating solution is stable. For motility on a substrate, curvature as in sporozoites leads to circular trajectories, hence bacteria necessarily have to find another solution to this fundamental constraint. We show that locally biasing the direction of the surface flow, which leads to helical surface flow fields, can accomplish this. Experimental observations suggest that the organization of the flow field enforced by bacteria might be more complex, featuring opposing helical tracks, which might be needed to ensure material transport in both directions. Prepatterning such surface flows yields comparable resulting global motion within our theory. In both cases, the initial stability of the rotational solution is mirrored by the mixed nature of the resulting motion.

Due to its general nature, our theory should also be useful to design synthetic microgliders. Moving microorganisms are a great inspiration for the design of autonomous nano- and microrobots, and many different designs have already been suggested (44). Gliding motility driven by surface flow of adhesins, like discussed here, has not been realized yet in this context, but the gliding motility of Mycoplasma mobile ghosts suggests that such designs might be possible (45). One interesting avenue in this context might be the use of DNA-origami, which has strongly advanced over the recent decade as an extremely versatile approach to engineer molecular machines (46). In particular, a synthetic DNA-glider has been engineered, in which DNA-nanotubes with different binding tracks, similar to the internal bacterial tracks discussed above, are propelled over a field of beating molecular motors (47). It has been noted before that shape design of such DNA-origami will determine its gliding motility (48), but for more autonomous control of these microgliders, the motors should be placed on the surface of the object itself, similar to the case of M. mobile.

In summary, our universal theory of gliding motility does not only explain many experimental observations, it also provides a useful reference to explain deviations from our predictions and to identify the ways in which microorganisms have managed to avoid the fundamental physical constraints geometry imposes on gliding motility. Moreover, it might be helpful for the design of synthetic microgliders.

Supplementary Material

Appendix 01 (PDF)

Movie S1.

Cylindrical cell in 3D environment. A spherocylindrical cell in a 3D environment is initialised in a translating solution, but with a small perturbation. It is transiting until arriving in the rotating solution, where it stops moving except for rotating around it’s axis. In blue the trajectory of the center, in red that of an off-axis point at the rear of the shape.

Movie S2.

Ellipsoidal shape in mixed motion. Ellipsoidal cell shape showing a stable mixed solution, that is simultaneous translation and rotation around its long axis. In blue the trajectory of the center, in red that of an off-axis point at the rear of the shape.

Movie S3.

Curved cell in 3D environment. A sufficiently curved cell shape prevents rotation to set in, thereby stabilizing the translational solution, which subsequently produces circular trajectories. In blue the trajectory of the center, in red that of an off-axis point at the rear of the shape.

Movie S4.

Curved cell in mixed motion. The curved cell shape displaying the stable mixed motion regime. For the curved shape, mixed motion means simultaneous translational motion along its axis, with associated rotation due to the curved shape (this alone would lead to a circular trajectory), but additionaly rotation around the direction of translations, such that the circle is bent up into a helix. In blue the trajectory of the center, in red that of an off-axis point at the rear of the shape.

Movie S5.

Cylindrical shape on substrate. A spherocylindrical cell positioned on a 2D substrate (e.g. a glass slide), initialised for translational motion, but with a small pertubation. As translational motion is unstable, the cell starts turning ever faster, until it is only rotating and the translation arrests. In blue the trajectory of the center, in red that of an off-axis point at the rear of the shape.

Movie S6.

Cylindrical shape on substrate with bias. Similar to S5, but now the surface flow has a small bias to flow from front to rear instead of tangentially around the cylinder. This bias is sufficient to shift stability from the rotating solution towards a mixed solution, i.e. the cell starts rotating as before, but maintains a translational component. In blue the trajectory of the center, in red that of an off-axis point at the rear of the shape.

Movie S7.

Cylindrical shape on substrate with prescribed two way helical surface flow. Similar to S5, but with a prescribed surface flow and coupling distribution over the spherocylinder as depicted in Fig. 7f, matching the patterns expected for gliding F. johsoniae.

Data, Materials, and Software Availability

All study data are included in the article and/or supporting information.