We consider the protein as a filament with monomeric subunits that bind to a membrane in an energetically favorable manner, such as burial of hydrophobic residues (Hussain et al., 2018). When a filament binds to a membrane, an energetic cost $E_{\mathrm{def}}({\mathrm{\ell }}_b)$ is associated to deformations which deviate from a position at mechanical equilibrium, while the free energy may be lowered by an amount $E_{\mathrm{int}}({\mathrm{\ell }}_b)$ due to interaction (Figure 2—figure supplement 1a). Both the deformation and interaction energies are expressed as functions of the bound filament length, ${\mathrm{\ell }}_b$, which is less than or equal to the total filament length, $L_f$. We wish to minimize the free energy due to filament binding, $\mathrm{\Delta }E=E_{\mathrm{def}}-E_{\mathrm{int}}$. If $\mathrm{\Delta }E({\mathrm{\ell }}_b)$ is negative, then it is energetically favorable for the filament to bind to the membrane along a length ${\mathrm{\ell }}_b$. We estimate $E_{\mathrm{int}}$ as

where $N_{\mathrm{int}}$ denotes the total number of membrane binding sites of the filament and ${\epsilon }_0$ denotes an independent and additive single binding site energy, which is given for MreB along with other parameter values in Supplementary file 1.

We assume that the binding sites are arranged linearly along the filament, and in particular, that the filament is not twisted. In this case, it suffices to account only for filament bending, and we may decompose the deformation energy $E_{\mathrm{def}}$ into the bending energy of the filament, $E_{\mathrm{bend}}$, and the deformation energy of the membrane, ${\displaystyle E_{\mathrm{m}\mathrm{e}\mathrm{m}}:E_{\mathrm{d}\mathrm{e}\mathrm{f}}=E_{\mathrm{b}\mathrm{e}\mathrm{n}\mathrm{d}}+E_{\mathrm{m}\mathrm{e}\mathrm{m}}}$. With notation as in the main text, we model the filament as a curved, cylindrical elastic rod with a circular cross-section of radius $r_f$ and curvature $1/R_s$, so that the elastic energy density per unit length of bending the filament from a curvature of $1/R_s$ to a curvature of $1/R$ is

where $Y$ is the elastic modulus of the filament and $B=\pi Y{r}_f^4/4$ is its flexural rigidity (Landau and Lifshitz, 1970). The resulting filament bending energy is $E_{\mathrm{bend}}={\epsilon }_{\mathrm{bend}}{\mathrm{\ell }}_b$. For simplicity, we have assumed the filament to be bent uniformly, but the case of a curvature which varies with position along the filament length can be considered similarly.

As stated in the main text, we assume an isotropic, fluid bilayer membrane, where there is no in-plane shear modulus and the only in-plane deformations are compressions and expansions. The membrane free energy assumes the form of Equation (1) in the main text. The mechanical energy required to bend a membrane from a surface $S_0$, with a mean curvature $H_0$, to a surface $S$ is then the difference of the corresponding free energies:

where the surface integrals of the Gaussian curvature are topological invariants by the Gauss-Bonnet theorem and therefore cancel in the difference, and the volume integral is understood to be a difference of the volumes in the deformed and undeformed states.

Minimizing $\mathrm{\Delta }E$ requires the minimization of $E_{\mathrm{mem}}$ given some value $1/R$ of the deformed filament curvature. To minimize $E_{\mathrm{mem}}$ in Equation (S3), we assume that the surface $S$ can be parameterized in the Monge gauge $h=h(x,y)$, where $(x,y)\in {ℝ}^2$, and furthermore that $|\nabla h|\ll 1$: this means that the membrane surface is not excessively curved or kinked. We assume the same for the undeformed surface $S_0$, which is parameterized by a function $h_0$ in the Monge gauge. In the case of binding to a cylindrical membrane, we may, for instance, take the undeformed surface to correspond to a cylinder with radius $R_{\mathrm{cell}}$:

In the Monge gauge, the mean curvature can be expanded as $H=\frac{1}{2}{\nabla }^{2}h+O[{(\nabla h)}^2]$, where the big-$O$ notation signifies $|H-\frac{1}{2}{\nabla }^{2}h|\le M{(\nabla h)}^2$ when ${\displaystyle 0<(\mathrm{\nabla }h{)}^2<\delta }$ for some positive numbers $\delta$ and $M$. The membrane bending energy in Equation (S3) can then be rewritten with the Laplacian, $\mathrm{\Delta }$, as

for some domain $\mathrm{\Omega }\subset {ℝ}^2$ of $h$ and $h_0$ not containing the domain $U$ of the filament surface (Figure 2—figure supplement 1a). Setting the first variation of $ℱ[h]$ to zero, we find that the equilibrium membrane shape is given by the solution of the shape equation

where ${\mathrm{\Delta }}^2$ is the biharmonic operator. Equation (S6) is subject to the Dirichlet boundary conditions

Here $\varphi$ and $\psi$ are indicator functions defined by their values on the boundary of ${\displaystyle \mathrm{\Omega }}$, ${\displaystyle \mathrm{\partial }\mathrm{\Omega }}$. In the case of a cylindrical membrane, for instance,

where, as above, $h_0$ parameterizes the undeformed surface $S_0$, $p_0(y)$ is a quadratic function describing the values of the filament height at the curve $\partial U$ parameterizing the binding region, and $C_0$ is the mean curvature of the filament along $\partial U$. Thus, the first condition in Equation (S7) comes from imposing continuity of membrane height with respect to the filament surface, while the second condition comes from imposing continuity of mean curvature. In live cells, we treat the periplasm as a rigid, undeformable body (Hussain et al., 2018), so that, for instance, $p_0(y)\ge h_0(x,y)$ for all $(x,y)\in \mathrm{\Omega }$ when $h_0$ assumes the form of Equation (S4) above. With the boundary conditions of Equation (S7), Equation (S6) can be conveniently decoupled as two Poisson equations, each with Dirichlet boundary conditions:

Since any solution to the Poisson equation with Dirichlet boundary conditions is unique, the decomposition above yields a unique solution for $h$.

As the foregoing considerations assume a fixed $\mathrm{\Omega }$ in determining h, the size of $\mathrm{\Omega }$ is an additional variable that must be considered. Since shape space is infinite-dimensional, determining $h$ for an arbitrarily large $\mathrm{\Omega }$ does not necessarily imply that the global minimum of $\mathrm{\Delta }E$ is achieved; neither does it necessarily determine the appropriate decay length of the indentation, since $h(x,y)=h_0(x,y)$ is generally not a solution to Equation (S9). It is possible that $\mathrm{\Delta }E$ may be minimized at a finite $\mathrm{\Omega }$. Due to the boundary conditions (Equation (S7) and (S8)), solutions of Equation (S9) over finite $\mathrm{\Omega }$ are continuous, with continuous mean curvatures, and could in fact be physically plausible. This subtlety can be addressed by choosing $\mathrm{\Omega }$ so that the numerically computed value of $\mathrm{\Delta }E$ is minimal among differently sized $\mathrm{\Omega }$, as discussed below.

Given values of $k_b,p,R_{\mathrm{cell}},C_0$, and the filament height along $\partial U$, we numerically solved Equation (S6) by individually solving the decoupled equations, Equation (S9), with a two-dimensional finite element Poisson equation solver (Figure 2—figure supplement 1b–c) (Burkardt, 2011). The value of $\mathrm{\Delta }E$ was then calculated numerically from the height function, $h$, by triangulating $h$ and extracting the mean curvature and enclosed volume of the resulting mesh using pre-existing MATLAB (Mathworks, Natick, MA) software (Mecklai, 2004; Kroon, 2014; Suresh, 2010). To find the energy-minimizing filament-membrane conformation, $\mathrm{\Delta }E$ was numerically computed while varying the deformed filament curvature, $1/R$, the size of $\mathrm{\Omega }$, and the size of, and the filament height in, $U$ (the size of $U$ corresponds to the bound length of the filament; consistent with the discussion below, we find that ${\mathrm{\ell }}_b=L_f$ in all cases of interest). As the simulation details above assume a perpendicular binding orientation relative to the long axis of a cylinder, for simplicity we model binding to deviatory angles by perpendicular binding to a different cell radius, $R$, where $\theta ={\cos }^{-1}(R/R_{\mathrm{cell}})$. These implementation details were used to generate Figure 2—figure supplement 1b–c and Figure 4—figure supplement 4.

In this section, following Hussain et al. (2018), we provide details of analytic calculations complementing the numerical calculations discussed above. For a cylindrical membrane whose radius is larger than the radius of curvature of a filament, it is energetically favorable for the filament to bind at an angle of $\theta ={90}^{\circ }$ relative to the long axis of the cell, as this orientation requires minimal bending of both the protein and the membrane. For deviatory angles, ${\displaystyle |\theta -{90}^{\circ }|>0^{\circ }}$, an effective correction to the cell radius $R_{\mathrm{cell}}$ is a multiplicative factor of $1/\cos \theta$, which makes binding less energetically favorable. The energetic penalty for MreB filaments binding at deviatory angles is approximately $40kT$ for a broad range of membrane pressures and depending on the membrane radius (below and Figure 2B of the main text). In general, higher membrane pressures make it more energetically favorable for the protein filament to bend, which minimizes the amount of volume displaced by the protein-membrane interaction (see below) (Hussain et al., 2018), while smaller membrane pressures make it more energetically favorable for the membrane to bend. Since a filament can always bend to conform to the membrane curvature, we see that large pressure differences across the membrane may enhance the energetic preference of an MreB filament for the perpendicular orientation.

We may summarize our results over a wide range of parameter values with an approximate phase diagram, assuming a cylindrical membrane (Figure 2C in the main text). We use the membrane pressure $p$ and the filament Young modulus $Y$, which varies with the bending energy $E_{\mathrm{bend}}$ of the filament, as order parameters. By considering only the volume and surface height displaced directly underneath the binding region of the membrane, the deformation energy of the binding region, $U$, is (Hussain et al., 2018)

where $b$ is the fraction of the filament cross-section needed to adhere to the membrane. The critical pressure at which the deformation is dominated by filament bending can be estimated as the value that sets $E_{\mathrm{def},U}$ to be minimal at $R=R_{\mathrm{cell}}$:

which, for the parameter values summarized in Supplementary file 1, estimates ${\displaystyle p^{\ast }\approx 20\text{ kPa}}$ (Hussain et al., 2018). This value of $p^{*}$ is smaller than estimates of the turgor pressures of both B. subtilis and E. coli (Supplementary file 1), suggesting that in vivo, MreB filaments always bend to adhere to the inner membrane. Equation (S10) is used under this assumption to generate the curves in Figure 2B of the main text. Furthermore, the linear dependence on ${\mathrm{\ell }}_b$ of the first term of Equation (S10) implies that MreB filaments bind fully along their lengths. If ${\displaystyle p<p^{\ast }}$, as is the case for vesicles, then both the membrane and the MreB filament can deform each other in a manner that minimizes the total energy, with the membrane shape determined by Equation (S6). For a large range of membrane pressures ${\displaystyle 0\le p\lesssim p^{\ast }}$, we find that bound MreB filaments induce membrane curvature, and for vesicles where the pressure difference across the membrane is vanishingly small, the shape equation also predicts that MreB filaments can grossly deform the membrane (Figure 2—figure supplement 1c), a prediction consistent with experimental observations (Salje et al., 2011; van den Ent et al., 2014). For a filament with fixed dimensions, both $E_{\mathrm{bend}}$ and the expression for $p^{*}$ in Equation (S11) are proportional to $Y$. Hence, $E_{\mathrm{bend}}\propto p^{*}$ as $Y$ increases, and this relation is shown as the diagonal line in Figure 2C of the main text.

Similarly, considering only the membrane curvature induced by the binding region $U$, the filament does not bend if ${\displaystyle E_{\mathrm{b}\mathrm{e}\mathrm{n}\mathrm{d}}+E>E_{\mathrm{i}\mathrm{n}\mathrm{t}}}$, where ${\displaystyle E=\pi b{k}_b{L}_f/r_f}$. If this inequality were satisfied, then the interaction energy $E_{\mathrm{int}}$ may be too small to justify membrane binding, which requires a combination of polymer and membrane bending. Note that, while we assume $b=1/6$ in this work, our results do not significantly change for different $b$, as shown in Figure 4 of (Hussain et al., 2018), and the regimes delineated in this section are summarized in Figure 2C of the main text.

Finally, we note that, for the parameter values summarized in Supplementary file 1, the prediction that it is energetically favorable for MreB filaments to align along the circumferential direction of a rod-like cell is robust in the case where the intrinsic curvature varies with position along the filament. In particular, a calculation based on Equation (S2) shows that this conclusion follows given that the filaments are, on average, more curved than the membrane. To see this, let ${\kappa }_s(\mathrm{\ell })$ denote the intrinsic filament curvature as a function of position along the filament length, $\mathrm{\ell }$, and $\kappa$ denote the deformed curvature. In live cells, $\kappa$ does not vary as a function of $\mathrm{\ell }$ because it is most energetically favorable for the filament to bend completely to match the ambient membrane curvature, as we have shown above. The total bending energy of the filament is then

When binding to an angle that deviates from the circumferential direction in a cylinder, the deformed curvature will be smaller: let $\kappa ={\kappa }_0-{\kappa }^{\prime }$, where ${\kappa }_0$ is the curvature along the circumferential direction of a cylinder and ${\kappa }^{\prime }\ge 0$ is a constant correction to ${\kappa }_0$ depending on the binding angle. Then, the difference in bending energies between binding in the direction of $\kappa$ as opposed the circumferential ${\kappa }_0$ direction is

which is larger than zero provided the filament is, on average, more curved than the cell (that is, the second term above is non-negative). Hence, because the binding orientation is robust, our model predictions will remain the same. Nevertheless, we note that cryo-EM experiments (Hussain et al., 2018; Salje et al., 2011; van den Ent et al., 2014) support the assumption of a uniformly bent MreB filament, and we have therefore focused on this case in the main text.

In the foregoing argument, we have assumed that $\kappa$ does not vary as a function of $\mathrm{\ell }$. The case in which $\kappa$ varies significantly as a function of $\mathrm{\ell }$, which is relevant to the geometry considered in Figure 3F of the main text, is more delicate. Note, however, that in this geometry $\kappa (\mathrm{\ell })$ can be explicitly calculated to show that MreB still orients in the largest principal direction. We consider, in particular,

where, at an axial coordinate $v$, $E_1$ is the bending energy of a (constantly curved) filament aligning in the axial direction (with principal curvature ${\kappa }_1$) and $E_2$ is the bending energy of a filament aligning in the circumferential direction (with principal curvature ${\kappa }_2$). Here the integration is over the axial coordinates corresponding to filament length, $[v-d,v+d]$. Figure 3—figure supplement 2c–d shows that, in this case, binding along the greatest principal direction generally still incurs the least bending energy.

Before discussing the implications of Equation (S16), we provide implementation details for the simulations and numerical calculations discussed below and in the main text. We numerically implemented the dynamics of Equation (S16) with Langevin simulations and verified our results with coarse-grained, continuum calculations corresponding to solutions of the corresponding Fokker-Planck equation. For the former, individual filaments were activated and deactivated as discussed in the main text, and their trajectories were simulated directly according to Equation (S16). The final positions of all filaments were pooled together to determine the corresponding filament concentrations. For the latter, we discretized $(u,v)$-space uniformly into $M_u\times M_v$ rectangular elements, computed the transition matrix corresponding to Equation (S16) and a given set of parameter values, and multiplied a vector corresponding to the filament number, $N_F$, to model the dynamics. Deactivation effects were modeled by multiplying the vector of filament numbers after each timestep by $e^{-\lambda }$, and filament concentrations were determined by dividing the vector of filament numbers by the corresponding surface area elements. Specific implementation details, such as geometric parameters, are summarized for each figure in this work in §3.

For a piecewise parametric surface, we allowed filaments to translocate between two subsurfaces while conserving the total step size, $L$, translocated at every timestep. For the piecewise geometries considered in this work, the filament coordinates were connected between subsurfaces in an obvious manner.

Consider a cylinder parameterized by $\mathbf{𝐫}=(a\cos u,a\sin u,av)$, where ${\displaystyle a>0}$, $u\in [0,2\pi ]$, $v\in [0,z]$, and $z$ is a large enough constant so that we do not consider filament translocation out of the cylinder. The nonvanishing components of the metric tensor are $g_{11}=g_{22}=a^2$. Hence, the step sizes in parameter space are identical at all timesteps $n$ and equal to ${\mathrm{\ell }}_n=\mathrm{\ell }=L/a$, and translocation occurs mainly along the $u$-axis. Assuming processive filament motion, we recover a Pearson-like random walk in a 2D plane with periodic boundary conditions in $u$, where the translocation angle satisfies ${\displaystyle {\theta }_n\sim N(0,{\sigma }^2)}$, and ${\chi }_n=\pm 1$ for all $n$. Correlated Pearson random walks of a similar form have been studied by Kareiva and Shigesada in the context of insect movement (Kareiva and Shigesada, 1983). Denoting by $U_N$ and $V_N$ the displacement along the $u$ and $v$ coordinate, respectively, after $N$ steps, with ${\displaystyle U_N=\sum _{i=1}^N\ell \cos {\theta }_i}$ and ${\displaystyle V_N=\sum _{i=1}^N\ell \sin {\theta }_i}$, we find that

in agreement with simulations of Equation (S16) for parameter values relevant to B. subtilis (§3.1 and Figure 3B of the main text).

Let $\mathbf{𝐫}=((R+a\sin u)\cos (av/R),(R+a\sin u)\sin (av/R),a\cos u)$, where $u\in [0,2\pi ]$, $v\in [0,R\mathrm{\Phi }/a]$, $\mathrm{\Phi }$ is the axial subtended angle, and ${\displaystyle 0<a<R}$. Here the nonvanishing elements of the metric tensor are $g_{11}=a^2$ and $g_{22}=a^2{(1+(a/R)\sin u)}^2$, and the step sizes depend on $u$. As the principal curvatures in the circumferential $u$-direction, $c_u$, and in the axial $v$-direction, $c_v$, satisfy ${\displaystyle c_u=1/a>c_v=(R+a\sin u{)}^{-1}\sin u}$, translocation occurs mainly in the direction of $u$. Due to the dependence of the step sizes on $u$, $⟨U_N⟩$, $⟨V_N⟩$, $\mathrm{Var}(U_N)$, and $\mathrm{Var}(V_N)$ may differ from that of a cylinder and are analytically difficult to calculate. We focus instead on determining the filament concentration, ${\displaystyle C_F}$, which, unlike the case of a cylinder, will be non-uniform in ${\displaystyle X}$.

For intuition, we consider the case of no translocation noise. Consider the probability $P_N(X;Y)$ of observing the filament at any point $X$ in $(u,v)$-space at a timestep $N$ given that it is initially at $Y$, and assume $p_{\mathrm{init}}(Y)$ to be proportional to $dA=a^2(1+(a/R)\sin u)dudv$. Since $dA$ depends on $u$, the probability $P_N(X)$ of observing the filament at a timestep $N$, averaged over initial positions, is not uniform. Nevertheless, considering the dynamics of activating and deactivating filaments as above, we find that, in the limit of continuous time, the expected number of filaments at a coordinate $X=(u,v)$ and time $t$ is

where $k^{\prime }=k{p}_{\mathrm{init}}/dA$ and $\nu$ is a speed corresponding to $U_N$ in the continuum limit. As $\lambda \to \mathrm{\infty }$, corresponding to the limit in which filaments are instantaneously deactivated, Equation (S21) predicts a number enhancement on the outer edge of the form $N_F(X,t)=(2a^2{k}^{\prime }/\lambda )(1+(a/R)\sin u)$, while as $\lambda \to 0$, corresponding to the limit in which filaments persist indefinitely, Equation (S21) predicts an approximate uniform distribution $2a^2{k}^{\prime }(t+\frac{a}{R\nu }\sin (t\nu )\sin u)\approx 2a^2{k}^{\prime }t$ in the limit of large $t$. While the concentration of filaments, $C_F$, is uniform over the surface in the former case, the latter case corresponds to the ‘washing-out’ of initial activation conditions and implies a larger value of $C_F$ on the inner edge. Namely, in the formal limit $\lambda \to 0$ followed by $t\to \mathrm{\infty }$, with $\lambda t\to 0$,

which is maximized at $u=-\pi /2$, corresponding to the inner edge of the torus. The filament concentration at the inner edge becomes larger than that of the outer edge by a factor of $(1+a/R)/(1-a/R)$, which depends only on the geometry of the torus. This effect may be interpreted as a ‘geometric focusing’ caused by both the filament number, $N_F$, becoming uniform in $(u,v)$-coordinates and variations in the area element $dA$ in $u$ or $v$. Even for a range of noises, including the limit of large noise ($\sigma \to \mathrm{\infty }$), simulations show that this description is valid and agrees with Equation (S22) (Figure 4—figure supplement 1a).

While the geometric focusing effect is clearly applicable to other geometries, particularly those involving bent or sinusoidal surfaces (Takeuchi et al., 2005; Renner et al., 2013), we examine whether it applies to ellipsoids and other geometries considered in this work below. We also note that this effect is different from filaments translocating along a direction of curvature until it is ‘attracted,’ as is the case for a spherocylinder (§2.5) or a surface with small wavelength undulations (Figure 3F of the main text and §2.8). The main difference is that the dynamics corresponding to geometric focusing is recurrent, and hence, due to spatial variations in the area element $dA$, leads to $C_F$ depending on geometric features such as in Equation (S22). In contrast, the dynamics corresponding to an attractor is not. For instance, in the absence of noise and the limit of large processivity, $C_F$ is nonzero over a toroidal surface and the relative localization is determined by Equation (S22), but $C_F$ vanishes at the hemispherical caps on a spherocylinder, as discussed later (§2.5).

Note that characteristic values of MreB persistence, as summarized in Supplementary file 2, suggest that MreB filaments lie in the $\lambda \approx 0$ regime. The observation of enhanced MreB concentration at the inner edge of toroidal cells is quantitatively consistent with the geometric focusing effect discussed in this section, as explained in the main text and prior work (Wong et al., 2017).

Finally, we note that similar arguments as that above describe the case of a helix, which can be parameterized by

where $u\in [0,2\pi ]$, $v\in [0,R\mathrm{\Phi }/a]$, $\mathrm{\Phi }$ determines the extended length of the helix, ${\displaystyle 0<a<R}$, and $\varphi$ is the helical pitch. Here the nonvanishing elements of the metric tensor are $g_{11}=a^2$, $g_{12}=g_{21}=\varphi a^3{(R\sqrt{R^2+{\varphi }^2})}^{-1}$, and ${\displaystyle g_{22}=a^2(2(R^2+{\varphi }^2{)}^2+a^2(R^2+2{\varphi }^2)}$ ${\displaystyle +aR(4(R^2+{\varphi }^2)\sin u-aR\cos 2u))(2R^2(R^2+{\varphi }^2){)}^{-1}}$, and the step sizes again depend on $u$. As the principal curvatures in the circumferential direction, $c_u$, and in the axial direction, $c_v$, satisfy ${\displaystyle c_u=1/a>c_v=(R^2+{\varphi }^2+aR\sin u{)}^{-1}R\sin u}$, translocation occurs mainly in the direction of $u$. Like the case of a torus, we expect filaments to become uniformly distributed across circumferential hoops in the limit of infinite processivity. The predicted value of $C_F$ in this case, $C_F\propto 1/dA={(a^2(R^2+{\varphi }^2+aR\sin u)dudv)}^{-1}R\sqrt{R^2+{\varphi }^2}$, is quantitatively consistent with numerical simulations and larger at the inner edge (Figure 4—figure supplement 1b–c).

Consider the surface $\mathbf{𝐫}=(a\sin u\cos v,a\sin u\sin v,b\cos u)$, where ${\displaystyle a,b>0}$, $u\in [0,\pi ]$, and $v\in [0,2\pi ]$. Here the nonvanishing components of the metric tensor are $g_{11}=(a^2+b^2+(a^2-b^2)\cos (2u))/2$ and $g_{22}=a^2{\sin }^{2}u$, and $dA=\frac{1}{\sqrt{2}}a\sin u\sqrt{a^2+b^2+(a^2-b^2)\cos (2u)}dudv$. When ${\displaystyle b>a}$, the principal curvatures in the circumferential $v$-direction, $c_v$, and in the axial $u$-direction, $c_u$, satisfy

with equality only at the poles ($u=0,\pi$). Hence, in the case of no noise, filaments always translocate in the direction of $v$, and similarly, the opposite is true when ${\displaystyle b<a}$.

As calculations involving a finite noise are analytically complex, we first consider the case of no noise. Here $U_N=0$ and $V_N^u=LN/(a\sin u)$, where $U_N$ is the displacement along the $u$-coordinate after $N$ steps and $V_N^u$ is the displacement along the $v$-coordinate after $N$ steps assuming that the filament remains on the curve of constant $u$. For an ensemble of filaments, considering similar activation and deactivation dynamics as above gives

where $k^{\prime }$ is the same as above and $\nu$ is a constant speed corresponding to $V_N=V_N^u\sin u$ in the continuum limit. Hence, for any value of $\lambda$, $C_F$ is a constant, in agreement with numerics (Figure 4—figure supplement 2).

We next consider the case of a finite noise. We first note that the area element $dA$ vanishes at the tips, and here the resulting value of $C_F$ would diverge if $N_F$ were nonzero. To compare filament concentrations over regions with differently sized area elements, it is convenient to define the average concentration on the (sub)surface $S$ as

which is different from a direct averaging of $C_F$ over area elements as considered for a torus (the latter has the form of a harmonic sum). Numerical calculations accounting for an ensemble of filaments with similar activation and deactivation dynamics as above show that, intriguingly, ${⟨C_F⟩}_S$ is enhanced at the ellipsoidal poles (${\displaystyle S=\{X:u<\pi /4\mathrm{o}\mathrm{r}u>3\pi /4\}}$) for a range of noises (Figure 4—figure supplement 2). Different from the case of zero noise, here the increased values of $C_F$ at the poles arise because filaments may randomly translocate to the poles and the area element is significantly smaller there than that away from the poles. Similar observations also hold in the limit of large noise, $\sigma \to \mathrm{\infty }$, and the limiting value of the ratio of ${⟨C_F⟩}_S$ at the tips versus the bulk is the ratio of the corresponding areas (Figure 4—figure supplement 2).

In the case of a finite noise, localization in a torus, a helix, and an ellipsoid arises because of $N_F$ becoming spatially homogeneous and spatial variations of the area elements. However, in the case of zero noise, we remark on a key difference between $C_F$ on a torus and an ellipsoid as follows. For a torus, spatially heterogeneous filament concentration arises due to the trajectories of individual filaments being closed orbits in the circumferential direction and the long persistence of filaments, which ‘washes out’ the initial distribution of filament position. For an ellipsoid, filaments do not move in the axial direction and the resultant filament concentration remains uniform on the surface, regardless of persistence. That a uniform distribution is maintained in the absence of noise on an ellipsoid is also different from the cases of a spherocylinder (§2.5) and a surface with small wavelength undulations (Figure 3F of the main text and §2.8), as discussed later. In general, $C_F$ would be uniform in the case of zero noise for any surface where the direction in which $dA$ varies does not coincide with the direction of filament translocation.

Consider the surface parameterized piecewise by ${\mathbf{𝐫}}_1=(a\cos u,a\sin u,av)$, where ${\displaystyle a>0}$, $u\in [0,2\pi ]$, $v\in [0,z]$ for some ${\displaystyle z>0}$, ${\mathbf{𝐫}}_2=(a\sin v\cos u,a\sin v\sin u,a\cos v+az)$, where $u\in [0,2\pi ]$ and $v\in [0,\pi /2]$, and ${\mathbf{𝐫}}_3={\mathbf{𝐫}}_2-(0,0,az)$, where $u\in [0,2\pi ]$ and $v\in [\pi /2,\pi ]$. In the absence of noise, filaments translocate along the $u$ axis in the cylinder and a random angle in the hemispheres. This could allow for translocation out of the caps and into the cylindrical body. Indeed, in the absence of noise, it is clear that any filament in the hemispherical caps will eventually translocate into and remain at the cylindrical rims.

For the spherocylindrical surface described above, general calculations in the case of a finite noise are analytically complex due to the irregular geometry. In the presence of noise, however, the depletion of filaments at the hemispherical caps can be supported numerically. For parameter values relevant to MreB in E. coli and B. subtilis (Supplementary file 2), numerical calculations show that ${⟨C_F⟩}_S$ is larger by a multiplicative factor of ∼2.0 when $S$ is the cylindrical bulk than that when $S$ is a hemispherical endcap (Figure 3C of the main text).

While filament movement depends on the principal curvatures, we wondered whether this implies that filament concentration always depend on the Gaussian curvature. Here we show that, for general surfaces, filament concentration is independent of Gaussian curvature under the dynamics considered in this work. Consider the parameterization $\mathbf{𝐫}=(\sin u,(1-\cos u)\cos u,v)$, where $u\in [-\pi ,\pi ]$, $v\in [0,z]$, and $z$ is a large enough constant so that we do not consider filament translocation out of the surface. Although the Gaussian curvature vanishes everywhere, any cross-section of the surface has regions of both negative and positive principal curvature, and the regions of positive principal curvature represent attractors to the filament dynamics: filaments translocating into such regions rarely translocate out.

While numerical simulations show that filaments are localized at regions where the largest principle direction coincides with the axial direction (Figure 4C in the main text and Figure 4—figure supplement 3a–b), analytical calculations are difficult to undertake due to the complicated geometry. Nevertheless, we may consider a similar 1D problem of a particle moving with velocity $\nu$ along a circular ring, which is parameterized without loss of generality by $u\in [0,2\pi ]$, and contains a single absorbing coordinate at $u=0$. Accounting for activation and deactivation dynamics as above, the filament number at $u=0$ at a time ${\displaystyle t>t^{\ast }}$ in the case of vanishing translocation noise ($\sigma =0$) and assuming ${\displaystyle \nu >2\pi \lambda }$, so that the flux described below is nonzero, can be written as

where $k^{\prime }$ is defined above, the term in parenthesis is the filament flux into $u=0$ at time $t_1$, and $t^{*}=2\pi /\nu$ represents the maximal time needed for any filament to become absorbed. Further assuming a constant value of $dA(u,v)=dA$ for simplicity, direct evaluation of Equation (S29) yields

which quantitatively describes the dependence of the filament number at the absorbing coordinate as a function of the processivity, which is determined by $\lambda$, and other parameters. Note that, as $\lambda \to 0$, corresponding to the case of infinite processivity, $N_F(u=0)$ is predicted to diverge, while as $\lambda \to \mathrm{\infty }$, corresponding to the case of zero processivity, $N_F(u=0)\to 0$. It is straightforward to generalize Equation (S30) to the case of several absorbing points and different geometries, provided that similar simplifying assumptions can be employed as above. Importantly, these calculations can be compared to numerical calculations for the geometry considered in this section. While the value of $N_F$ at attracting regions may generally depend on the geometry of such regions, the dependence of $N_F$ on the processivity, $\lambda$, can be conveniently explored by defining the localization ratio, $\rho$, as the ratio between filament numbers at a certain value of $\lambda$ compared to that at $\lambda /2$:

Defining $\rho$ analogously for the geometry considered in this section, we find that numerical calculations of $\rho$ for the geometry considered here across a range of $\sigma$ are consistent with the theoretical prediction for $\rho$ based on the simplified model considered in this paragraph (Figure 4—figure supplement 3c). Hence, we conclude that Equation (S30) captures the dependence of $N_F$ on $\lambda$ for more general geometries.

Building on work that has examined MreB motion in cells with membrane bulges (Hussain et al., 2018), here and below we consider bulges of positive Gaussian curvature on a cylindrical surface. We consider a class of cylinders with bulges parameterized piecewise by ${\mathbf{𝐫}}_1=(v,\cos u,\sin u)$, where $u\in [0,2\pi ]$ and $v\in [0,z]$ for some ${\displaystyle z>0}$ and $\sqrt{{(u-\pi /2)}^2+{(v-z/2)}^2}<b$ and ${\mathbf{𝐫}}_2=(b\sin v\cos u+z/2,b\sin v\sin u,sc\cos v+1)$ where $u\in [0,2\pi ]$ and $v\in [0,\pi /2]$ otherwise. We suppose ${\displaystyle b,c>0}$ and ${\displaystyle b<z}$, so that the intersection area constitutes a small fraction of the cylindrical body. Here $s=1$ if the bulge protrudes outward and $s=-1$ if the bulge protrudes inward. For this class of parametric surfaces, three general cases of outward bulges ($s=1$) can be classified depending on whether $c=b$, ${\displaystyle c<b}$, or ${\displaystyle c>b}$. Upon computing the principal curvatures in each case, we find that the three cases correspond respectively to random translocation ($c=b$), in which case the translocation direction is random; polar translocation (${\displaystyle c<b}$), in which case translocation occurs predominantly along the bulge $v$-coordinate; and circumferential translocation (${\displaystyle c>b}$), in which case translocation occurs predominantly along the bulge $u$-coordinate (Figure 3—figure supplement 1a). We note that inward bulges of positive Gaussian curvature ($s=-1$) exhibit the same behavior, with polar translocation when ${\displaystyle c>b}$ and circumferential translocation when ${\displaystyle c<b}$.

Figure 4B of the main text illustrates filament dynamics for both random and circumferential translocation where $b=c=0.5$ (spherical bulge) and $b=0.5,c=1$ (ellipsoidal bulge), in both the cases of zero and infinite processivity. We focus here on the latter geometry as the motion there is consistent with experiments (Hussain et al., 2018). When there is no noise, considering similar activation and deactivation dynamics as above shows that, similar to a spherocylinder (§2.5), the bulge attracts filaments in the case of small $\lambda$ or large processivity. As simulations demonstrate, this is also true in the case of a finite noise, where we find the bulge to contain larger numbers of filaments relative to the case of a uniform distribution on the surface (Figure 4B in the main text). For characteristic parameter values relevant to B. subtilis MreB, we find that ${⟨C_F⟩}_S$ is increased at the bulge (Figure 3E of the main text) and substantial localization may occur at the bulge neck, consistent with previous experiments (Figure 3—figure supplement 1b).

Previous work has examined the correlation of MreB concentration with subcellular-scale shape fluctuations in E. coli cells (Ursell et al., 2014). Based on the weak correlation observed between outer and inner contour curvatures in Ursell et al. (2014), the authors determined that the experimentally observed MreB enrichment was not caused by bending modes: in a circular torus, for instance, the correlation between outer and inner contour curvatures should be strictly negatively correlated. The authors concluded that short length-scale, high magnitude fluctuations dominate experimental observations of cell shape.

To probe such a geometry, we consider the parameterization $\mathbf{𝐫}=((a+c\sin (Pv))\sin u,(a+c\sin (Pv))\cos u,v)$, for $u\in [0,2\pi ]$ and $v\in [0,2\pi ]$. Here $a$ denotes an average cylinder radius, we assume that ${\displaystyle 0<c\ll a}$, and ${\displaystyle P>0}$ is a variable controlling the number of periods. For large wavelength undulations, $P$ is less than, or on the order of, unity. However, the Gaussian and mean curvatures are uncorrelated in this case, inconsistent with the positive correlation observed in Ursell et al., 2014 for cells growing in sinusoidally-shaped chambers, and the ranges of Gaussian and mean curvatures are significantly smaller than those measured for wild-type, unconfined, sinusoidally-confined, thin, and wide cells in different experiments (Ursell et al., 2014; Shi et al., 2017; Bratton et al., 2018) (Figure 3—figure supplement 2a and Figure 3G of the main text). We therefore consider a geometry with short wavelength undulations, for which ${\displaystyle P\ge 1}$ (Figure 3—figure supplement 2b). Numerical results for principal curvature-dependent translocation on this geometry, which is consistent with the predicted filament binding orientation (Figure 3—figure supplement 2c–d), are presented in Figure 3G in the main text.

Prior experiments have shown that the variation in MreB trajectory direction is width-dependent, suggesting that the translocation noise may depend on the difference of principal curvatures, $\mathrm{\Delta }c$ (Hussain et al., 2018). As discussed in the main text, we may model this observation by letting $\sigma$ vary with $\mathrm{\Delta }c$: for simplicity, we set

(Figure 3—figure supplement 3a), but note that more complicated functional dependencies, such as a quadratic dependence of the form $\sigma =\beta {(\mathrm{\Delta }c)}^{-2}$, do not significantly change our results (Figure 3—figure supplement 3b). Unless otherwise indicated (see §3), all simulations and calculations in this work pertaining to MreB assume Equation (S32), and we further note that, for the parameter values relevant to MreB translocation in E. coli and B. subtilis summarized in Supplementary file 2, taking a constant value of $\sigma =0.3$ also does not significantly change our results. Similarly, we verify that localization arises even for vary large step sizes ($L=2\mu \text{m}$), in which case MreB filaments realign infrequently (Figure 3—figure supplement 3b).

Recent work has shown that regions of negative Gaussian curvature can allow twisted filaments to bind with low elastic energy (Quint et al., 2016). In another study, Wang and Wingreen assumed that MreB assembles into bundles with significantly larger flexural rigidity than that of filaments considered in this work (Wang and Wingreen, 2013). In this section, we examine how our model predictions for MreB localization in E. coli change over ranges of three parameters: (1) the intrinsic filament twist, ${\omega }_0$, (2) the coupling, $\gamma$, of filament activation to Gaussian curvature, and (3) the filament flexural rigidity, $B$.

We first note that $\gamma$ may vary independently of ${\omega }_0$. Quantitatively, the rate of filament activation may not only depend on the parameters of the twist, but also on other cellular parameters (Wong et al., 2017). We assume that the filament activation rate per unit area, $k$, varies with $\gamma$ as

where $k_0$ and $\gamma$ are constants and $G(u,v)$ is the Gaussian curvature at the parametric coordinate $(u,v)$. When $\gamma =0$, we recover our original assumption that the activation rate per unit area is constant.

We now explore the effects of twist and Gaussian curvature-dependent activation on filament concentration. For simplicity, we consider a torus (Figure 4—figure supplement 1a), for which $G=\sin u{[a(R+a\sin u)]}^{-1}$ (c.f. §2.3), but show later in Figure 3—figure supplement 3b that the results for the undulating geometry of Figure 3F of the main text also remain qualitatively similar. Due to twist, MreB filaments may move in a direction which deviates from the direction of largest curvature. In particular, while further work should verify the robustness of this equation for nonzero turgor pressures and intrinsic filament curvatures, Equation (12) of Quint et al. (2016),

predicts the angular deviation from the largest principal direction in a cylinder, ${\theta }_0$, due to twist in the limit that MreB filaments are fully bound to the membrane. Here $B$ is the flexural rigidity of a filament, $K$ is the elastic twist stiffness, $a$ is the cell radius, and ${\omega }_0$ is the intrinsic helical twist. Thus, upon knowing the values of $k_0$, ${\omega }_0$, $\gamma$, and $B$, we may compute $k$ and ${\theta }_0$ using Equation (S33) and (S34), from which we can determine $C_F$ via simulations similar to those above. Doing so for a large parameter range which includes characteristic values of $B$ (Supplementary file 1) and theoretically hypothesized values for MreB of $K=2000kT\cdot \text{nm}$ and ${\omega }_{0}a=1$ to $5$ (Quint et al., 2016) reveals the final ratio of filament concentrations to be quantitatively similar in all cases (Figure 4—figure supplement 4a–f). We note, in particular, that the effect of filament twist alone is small in all cases: this is because the biasing of translocation angles due to twist, as predicted by Equation (S34), is irrelevant for the toroidal geometry, where translocating along deviatory angles still traces out hoops (c.f. §2.3). In these simulations, to illustrate the range of translocation behavior we have assumed that binding is always energetically favorable; however, there will be an energetic cost of unwinding of the form

where $\omega$ is the bound filament twist (Quint et al., 2016). For the parameter values considered in this work (Supplementary file 1), binding becomes energetically unfavorable at large twists (${\displaystyle {\omega }_{0}a\gtrsim 100}$), for which ${\theta }_0\approx \pi /4$.

Additionally, our previous measurements of MreB fluorescence in bent E. coli cells confined to toroidal microchambers (which may be modeled as torii with geometric parameters $a=1$ and $R=10$) shows MreB concentration to be enhanced at the inner edges by a factor of approximately $1.1$ (Wong et al., 2017). This suggests the strength of Gaussian curvature coupling for MreB, if indeed such coupling does exist, to be small, as shown in Figure 4—figure supplement 4g–h. As we have demonstrated previously (Wong et al., 2017), the small observed enhancement is consistent with processivity alone. Furthermore, as mentioned above, modeling characteristic parameter values of both filament twist and Gaussian curvature-dependent activation still results in localization for geometries other than a torus, such as that considered in Figure 4F of the main text (Figure 3—figure supplement 3b).

Finally, we were interested to determine if our model predictions were robust to variation in the filament flexural rigidity, which for the parameter values in Supplementary file 1 has a value of $B\approx 1.65\times {10}^{-25}\text{J}\cdot \text{m}$. To explore this for a torus, we repeated the foregoing dynamical simulations across a range of flexural rigidities. We first considered a ten-fold smaller value of $B=1.0\times {10}^{-26}\text{J}\cdot \text{m}$, which, by Equation (S10), still predicts (1) the filament to bend to conform to the membrane and (2) the depth of the potential well corresponding to Figure 2B of the main text to be approximately $3kT$, a value that may be large enough to be robust to thermal noise and other sources of stochasticity (Figure 4—figure supplement 4k). We next considered a 100-fold larger value of ${\displaystyle B=1.5\times {10}^{-23}\text{ J}\cdot \text{m}}$, approximately the flexural rigidity of a thick bundle with ${\displaystyle r_f=10\mathrm{n}\mathrm{m}}$ (Wang and Wingreen, 2013). For this value of $B$ and $r_f$, no twist, and the remaining parameters summarized in Supplementary file 1, Equation (S11) estimates the critical pressure to be ${\displaystyle p^{\ast }\approx 5\mathrm{a}\mathrm{t}\mathrm{m}}$, a value which is larger than characteristic estimates of turgor in E. coli. As the corresponding value of $E_{\mathrm{bend}}$ is less than $E_{\mathrm{int}}$, we anticipate that both the membrane and the bundle may bend (see Figure 2C of the main text). To show that it remains energetically favorable for thick bundles to bind to membranes and that the binding orientation along the largest principal direction is robust, we numerically solved the shape equation (Equation (S6)) and found that the free energy change due to binding, $\mathrm{\Delta }E$, decreases with the size of the domain $\mathrm{\Omega }$ and, for a given $\mathrm{\Omega }$, is minimal when the membrane bends to conform to the bundle (Figure 4—figure supplement 4k). In the limit $\mathrm{\Omega }\to U$, the Monge gauge assumption underlying Equation (S6) becomes invalid; nevertheless, $\mathrm{\Delta }E$ tends to the analytical expression of Equation (S10) under the condition that the minimizer, $R$, is close to the intrinsic radius of curvature of the bundle, $R_s$. In both this limit and the simulations of Figure 4—figure supplement 4k, binding remains energetically favorable and the binding orientation remains robust even for thick filaments, suggesting that translocating along directions of largest principal curvature remains relevant. Figure 4—figure supplement 4f,i–j shows simulation results for both values of $B$ compared to the value ($1.65\times {10}^{-25}\text{J}\cdot \text{m}$) assumed in this work. We find that, in all cases, the model predictions remain quantitatively similar.

Here $a=1$ and $z=100$. Langevin simulations to generate trajectories were undertaken with ${10}^5$ activated filaments, $L=0.4$, $\sigma =0.3$, and $N$ determined by the number of steps needed to translocate one hoop. Filaments were activated at the center ($v=0$) so that none of them translocated beyond the range specified by $z$.

Here $a=1$ and $z=4$. Langevin simulations to generate a representative trajectory were undertaken with a single activated filament and the parameters relevant to E. coli summarized in Supplementary file 2, except $N=500$. Note that we use the linear relation $\sigma =\alpha {(\mathrm{\Delta }c)}^{-1}$, where $\mathrm{\Delta }c$ is the difference of principal curvatures and the value of $\alpha$ is provided in Supplementary file 2. Numerical calculations for ensemble dynamics were undertaken with the parameters relevant to E. coli summarized in Supplementary file 2. Each subsurface is discretized into $60\times 60$ bins.