Modeling Curvature-Dependent Subcellular Localization of the Small Sporulation Protein SpoVM in Bacillus subtilis

Recent in vivo experiments suggest that in the bacterium, Bacillus subtilis, the cue for the localization of the small sporulation protein, SpoVM, an essential factor in spore coat formation, is curvature of the bacterial plasma membrane. In vitro measurements of SpoVM adsorption to vesicles of varying sizes also find high sensitivity of adsorption to vesicle radius. This curvature-dependent adsorption is puzzling given the orders of magnitude difference in length scale between an individual protein and the radius of curvature of the cell or vesicle, suggesting protein clustering on the membrane. Here we develop a minimal model to study the relationship between curvature-dependent membrane adsorption and clustering of SpoVM. Based on our analysis, we hypothesize that the radius dependence of SpoVM adsorption observed in vitro is governed primarily by membrane tension, while for in-vivo localization of SpoVM, we propose a highly sensitive mechanism for curvature sensing based on the formation of macroscopic protein clusters on the membrane.


Absorption profiles in the absence of repulsion
In this section we will demonstrate the necessity of including repulsive interactions in order to fit the experimental data for membrane absorption versus SpoVM concentration (see Fig. 4, main text). In the absence of repulsive interactions, with increasing SpoVM concentration we find an abrupt transition to vesicles saturated with SpoVM. In the absence of repulsion the only relevant energetics governing SpoVM absorption are nearest neighbor attraction as well as the binding energy of individual proteins on the membrane. With this in mind we can reduce the problem to an Ising model. The energetics of SpoVM on the vesicle can be given by where, < i, j > indicates a sum over nearest neighbor lattice sites on the vesicle. σ i is assumed to be 1 if a lattice site is occupied and 0 otherwise. Let v be the total number of sites on the vesicle and n be the average occupancy per site of the vesicle. In the standard mean field approximation [1], the partition function of the system can be written as where β = 1 k B T , T is the temperature, µ is the chemical potential and z is the number of nearest-neighbor sites to each site on the vesicle. Since the average vesicle occupancy is given by n = 1 v d ln Z dµ we obtain the following relation n = e β( nz+µ) 1 + e β( nz+µ) .
(S3) Figure S1: Absorption versus protein molar concentration in solution in the case of zero repulsion.
Chemical potential is related to the concentration of proteins in the solution C s by the relationship e µ = βC s . Plugging this in the equation above we get n = e β nz C s 1 + e β nz C s (S4) The plot of n vs C s for β z = 4 is shown in Fig. S1. While the plot appears to be sensitive to the choice of β z, we note that exp(β nz) multiplies C s in the equation above, and thus can absorbed into C s . Thus the choice of β z does not affect the nature of the curve. We find that above a critical concentration the absorption of proteins on the vesicle spikes, implying a phase transition from SpoVM existing as monomers on the smooth surface of the vesicle to SpoVM forming one giant cluster. In order to smoothen this abrupt rise in absorption for increasing SpoVM concentration, repulsive interactions have to be included.

Technical details of the full model derivation
The entropy is where n v = k kn k . Using the sterling approximation ln N ! = N ln N − N , the above equation becomes where n V = k km k . The free energy is given by In order to obtain the equilibrium distribution of cluster sizes, we minimize F and set ∂F ∂m k = 0. This gives us implying Defining n * s = ns Ls , this can be written as If m k << L v and n * s << 1, we have and hence, Thus SpoVM absorption can be expressed as eq. (S15) becomes where c s = αn * s , c 0 = e −( nn+eb)/kB T α, with α being a constant. If we define the normalized concentration of proteins in the solution, C s = c s /c 0 and the normalized concentration of proteins bound to the vesicle C v = n v /(L v e − nn/kB T ), and e r = r /k B T we can write Let us evaluate the value of k = k * corresponding to the term with the largest contribution to the above sum. For this purpose, we have to maximize by setting its derivative with respect to k to zero. This yields We can approximate the largest term contributing to the absorption C v for large k * : implying

Stoichiometric clustering
We now consider the case when the membrane bound SpoVM forms complexes of a specific size. While we have no strong reason to suspect the presence of a stoichiometric complex, we consider this case for completeness. On the membrane we might expect to have monomers and n-mers where n is around 3 based on the date presented in Fig 1 and 4 (main text). In line with our discussion for the non-stoichiometric case, we assume that the repulsive energy of the n-mers have a radius dependence given by the form e r = c 1 + c 2 r. The absorption of SpoVM on the vesicle in this case can be written as In Figs. S2 and S3, we plot absorption as a function of vesicle radius and of concentration in solution in this case. While the data is still reasonably well fit by the stoichiometric model, results from the stoichiometric model are expected to deviate sharply from our clustering model if VM concentration is further increased or vesicle radius is further reduced. In Fig. S4 we plot the average cluster size (defined as the total number of membraneassociated VM divided by the total number of clusters) as a function of VM concentration. Again, we see a sharp difference between the two models at higher values of concentration.

Protein absorption and clustering for finite-range repulsion
As discussed in the main text, let us assume that repulsion between membranebound SpoVM has a finite range with a sharp cutoff (resembling a stepfunction). In this case, repulsion energy of a cluster scales as cluster size squared, (k 2 ), for clusters smaller than a critical size k * , but increases linearly with cluster size for larger clusters. For a linear cluster, the total energy for k > k * can be written as Hence the total energy associated with membrane bound SpoVM becomes  Figure S5: Average cluster size versus concentration of proteins on the vesicle for the finite-range versus infinite-range repulsion for e r = .5802 (r = .75 µm from our infinite-range repulsion model that best fits the data). The finite-range repulsion is assumed to be a step function with zero strength for proteins farther apart than fifth nearest neighbor.
Setting e r = r /k B T , due to the finite-range repulsion Eq (S18) becomes, We can easily see that as x approaches 1, C v diverges implying that SpoVM absorption on the membrane reaches maximum capacity. Thus, as discussed in the text, in the case where we have two membrane regions with slightly different curvatures and, consequently, binding energies, the values of c 0 and hence of x will be slightly different in the two regions. Thus, as x approaches 1 in the region with the higher curvature corresponding to a slightly higher binding energy, the value of x remains smaller than 1 in the other region, leading to pronounced SpoVM localization.