Bacterial microcompartments are large, roughly icosahedral shells that assemble around enzymes and reactants involved in certain metabolic pathways in bacteria. Motivated by microcompartment assembly, we use coarse-grained computational and theoretical modeling to study the factors that control the size and morphology of a protein shell assembling around hundreds to thousands of molecules. We perform dynamical simulations of shell assembly in the presence and absence of cargo over a range of interaction strengths, subunit and cargo stoichiometries, and the shell spontaneous curvature. Depending on these parameters, we find that the presence of a cargo can either increase or decrease the size of a shell relative to its intrinsic spontaneous curvature, as seen in recent experiments. These features are controlled by a balance of kinetic and thermodynamic effects, and the shell size is assembly pathway dependent. We discuss implications of these results for synthetic biology efforts to target new enzymes to microcompartment interiors.
While it has long been recognized that membrane-bound organelles organize the cytoplasm of eukaryotes, it is now evident that protein-based compartments play a similar role in many organisms. For example, bacterial microcompartments (BMCs) are icosahedral proteinaceous organelles that assemble around enzymes and reactants to compartmentalize certain metabolic pathways [1–10]. BMCs are found in at least 20% of bacterial species [2,11,12], where they enable functions such as growth, pathogenesis, and carbon fixation [1,10,13–16]. Other protein shells act as compartments in bacteria and archea, such as encapsulins [17] and gas vesicles [17,18], and even in eukaryotes (e. g. vault particles [19]). Understanding the factors that control the assembly of BMCs and other protein-based organelles is a fundamental aspect of cell biology. From a synthetic biology perspective, understanding factors that control packaging of the interior cargo will allow reengineering BMCs as nanocompartments that encapsulate a programmable set of enzymes, to introduce new or improved metabolic pathways into bacteria or other organisms (e. g. [10,20–29]) ]. More broadly, understanding how the properties of a cargo affect the assembly of its encapsulating container is important for drug delivery and nanomaterials applications. Despite atomic resolution structures of BMC shell proteins [1,10,30,31], the factors that control the size and morphology of assembled shells remain incompletely understood. BMCs are large and polydisperse (40-600 nm diameter), with a roughly icosahedral protein shell surrounding up to thousands of copies of enzymes [1,7–9,30,32,33]. For example, the best studied BMC is the carboxysome, which encapsulates RuBisCO and carbonic anhydrase to facilitate carbon fixation in cyanobacteria [1,30,32,34]. BMC shells assemble from multiple paralogous protein species, which respectively form homo-pentameric, homo-hexameric, and pseudo-hexameric (homo-trimeric) oligomers [1,30,31]. Sutter et al. [31] recently obtained an atomic-resolution structure of a complete BMC shell in a recombinant system that assembles small (40 nm) empty shells (containing no cargo). The structure follows the geometric principles of icosahedral virus capsids, exhibiting T = 9 icosahedral symmetry in the Caspar-Klug nomenclature [35,36] (meaning there are 9 proteins in the asymmetric unit). The pentamers, hexamers, and pseudo-hexamers occupy different local symmetry environments. Although the Sutter et al. [31] structure marks a major advance in understanding microcompartment architectures, it is uncertain how this construction principle extends to natural microcompartments, which are large (100-600 nm), polydisperse, and lack perfect icosahedral symmetry. Moreover, the effect of cargo on BMC shell size is hard to interpret from experiments. In some BMC systems, empty shells are smaller and more monodisperse than full shells [23,28,31,37], whereas in other systems empty shells are larger than full ones [38]. Thus, the cargo may increase or decrease shell size. The encapsulated cargo can also affect BMC assembly pathways. Microscopy experiments showed that β-carboxysomes (which encapsulate form 1B RuBisCO) undergo two-step assembly: first the enzymes coalesce into a ‘procarboxysome’, then shells assemble on and bud from the procarboxysome [39,40]. In contrast, electron micrographs suggest that α-carboxysomes (another type of carboxysome that encapsulates form 1A RuBisCO) assemble in one step, with simultaneous shell assembly and cargo coalescence [33,41]. Our recent computational study [42] suggested that the assembly pathway depends on the affinity between cargo molecules. However, that study was restricted to a single shell size, and thus could not investigate correlations between assembly pathway and shell size. Numerous modeling studies have identified factors controlling the thermodynamic stability [43–45] or dynamical formation [46–54] of empty icosahedral shells with different sizes. For example, Wagner and Zandi showed that icosahedral shells can form when subunits sequentially and irreversibly add to a growing shell at positions which globally minimize the elastic energy, with the preferred shell size determined by the interplay of elastic moduli and protein spontaneous curvature. Several studies have also investigated the effect of templating by an encapsulated nanoparticle or RNA molecule on preferred shell size [50,55–57]. However, the many-molecule cargo of a microcompartment is topologically different from a nucleic acid or nanoparticle, and does not template for a specific curvature or shell size. Rotskoff and Geissler recently proposed that microcompartment size is determined by kinetic effects arising from templating by the cargo [58]. Using an elegant Monte Carlo (MC) algorithm they showed that proteins without spontaneous curvature, which form polydisperse aggregates in the absence of cargo, can form kinetically trapped closed shells around a cargo globule. However, there are reasons to question the universality of this mechanism for microcompartment size control. Firstly, several recombinant BMC systems form small, monodisperse empty shells [23,28,31,37], suggesting that the shell proteins have a non-zero spontaneous curvature even without cargo templating. Secondly, when Cameron et al. [39] overexpressed RuBisCO to form ‘supersized’ procarboxysomes, carboxysome shells encapsulated only part of the complex, suggesting that there is a maximum radius of curvature that can be accommodated by the shell proteins. Thirdly, the kinetic mechanism is restricted to systems in which rates of shell association vastly exceed cargo coalescence rates, a condition which may not apply in biological microcompartment systems. Thus, despite this and other recent simulation studies of microcompartments [42,58,59], the factors which control BMC size and amount of encapsulated cargo remain unclear. In this article we use equilibrium calculations and Brownian dynamics (BD) simulations on a minimal model to identify the factors that control the size of a microcompartment shell. Although computationally more expensive than the MC algorithm of Ref. [58], BD better describes cooperative cargo-shell motions and thus allows for any type of assembly pathway. Using this capability, we explore the effect of cargo on shell size and morphology over a range of parameters leading to one-step or two-step assembly pathways. To understand the interplay between shell curvature and cargo templating, we consider two limits of shell protein interaction geometries: zero spontaneous curvature and high spontaneous curvature, which respectively form flat sheets or small icosahedral shells in the absence of cargo. Our calculations find that the presence of cargo can increase or decrease shell size, depending on the stoichiometry of cargo and shell proteins, and the protein spontaneous curvature. For shell proteins with high spontaneous curvature, we observe a strong correlation between assembly pathway and shell size, with two-step assembly leading to larger shells than single-step pathways or empty shell assembly. This result is consistent with the fact that β-carboxysomes tend to be larger than α-carboxysomes. For shell proteins with zero spontaneous curvature, we find that introducing cargo can result in a well-defined shell size through several mechanisms, including the kinetic mechanism of Ref. [58] and the ‘finite-pool’ effect due to a limited number of cargo particles available within the cell. However, spontaneous curvature of the shell proteins allows for robust shell formation over a wider range of parameter space.
We simulated assembly dynamics using the Langevin dynamics algorithm in HOOMD (which uses GPUs to efficiently simulate dynamics [81]), and periodic boundary conditions to represent a bulk system. The subunits are modeled as rigid bodies [82]. Each simulation was performed in the NVT ensemble, using a set of fundamental units [83] with σ0 defined as the circumradius of the pentagonal subunit (the cargo diameter is also set to σ0), and energies given in units of the thermal energy, kBT. The simulation time step was 0. 005 in dimensionless time units, and we performed 3 × 106 timesteps in each simulation unless mentioned otherwise. Initial conditions. We considered two types of initial conditions. Except where stated otherwise, simulations started from the ‘homogeneous’ initial condition, in which subunits and (if present) cargo were initialized with random positions and orientations, excluding high-energy overlaps. In the ‘pre-equilibrated globule’ initial condition, we first initialized cargo particles with random positions (excluding high-energy overlaps), and performed 105 simulation timesteps to equilibrate the cargo particles. Shell subunits were then added to the simulation box with random positions and orientations, excluding high-energy overlaps. Systems. We simulated several systems as follows. For shell subunits with spontaneous curvature we set pentamer-hexamer and hexamer-hexamer angles consistent with the T = 3 geometry (see Estimating the shell bending modulus in section S2 Text), and we set εangle = 0. 5. We first performed a set of empty-shell assembly simulations, with 360 hexamers, and varying number of pentamers, in a cubic box with side length 60σ0, with εHH = 2. 6kBT (the smallest interaction strength for which nucleation occurred). These simulations were performed for 107 timesteps to obtain sufficient statistics at low pentamer concentrations despite nucleation being rare. For cargo encapsulation by subunits with spontaneous curvature, we simulated 2060 cargo particles, 180 pentamers, and 360 hexamers in a cubic box with side length 60σ0. Other parameters were the same as for the empty-shell simulations, except that we varied εPH, εSC, and εSC as described in the main text. All simulations with spontaneous curvature used εPH ≥ 1. 3εHH to ensure that the shells with the T = 3 geometry (or asymmetric shells with similar sizes) were favored in the absence of cargo. We note that our results generalize to other ranges of shell interaction parameters, but this choice distinguishes effects due to cargo from those due to changes in the inherent preferred shell geometry. Simulations with strong cargo-cargo and cargo-shell interactions (εCC ≥ 1. 55 and εSC < 8. 75) required a long timescale for pentamers to fill pentameric vacancies in the hexamer shell (discussed in Results). To observe pentamer adsorption, these simulations were run for up to 9 × 106 simulation timesteps. For simulations of ‘flat’ subunits (with no spontaneous curvature), we considered a range of system sizes at fixed steady state cargo chemical potential, with the number of cargo particles varying from 409 to 3275, and the box side length varying from 35σ0 to 70σ0. Since these were NVT simulations, we ensured that the final hexamer chemical potential was the same at each system size by setting the number of hexamers so that the concentration of free hexamers remaining after assembly of a complete shell was constant (10−3 subunits/σ 0 3). The resulting number of hexamers varied from 109 to 581 in boxes with side lengths 35σ0 to 70σ0. The assembly outcomes were unchanged if instead we kept the total hexamer subunit concentration the same across all simulations. For each of these system sizes we performed simulations over a range of εangle to identify the maximum value of κs at which assembly of a complete shell could occur. Simulations were stopped upon completion of a shell or after the maximum simulation time tmax with tmax = 3 × 106 timesteps for boxes with side length ≤ 55σ0 and tmax = 8 × 106 for boxes with side length ≥ 55σ0. The maximum simulation time was increased for large system sizes because the minimum time required for assembly of a complete shell increases linearly with the shell size [84]. To estimate the relationship between the shell bending modulus κs and the parameter εangle we performed additional simulations, in which we measured the total interaction energy of completely assembled shells as a function of εangle (see ‘Estimating the shell bending modulus’ in section S2 Text). Sample sizes. For simulations of shells with spontaneous curvature, we performed a minimum of 10 independent trials at each parameter set. To enable satisfactory statistics on shell size and morphology for parameter sets that result in at most one complete shell in the simulation box 3, we performed additional trials such that at least 10 complete shells were simulated. For flat subunits (Fig 1F and 1G), we identified the maximum εangle for which a complete shell forms at each system size as follows. We first performed independent simulations over a range of εangle values, separated by increments in εangle of 0. 02 for systems with box side length ≤ 55σ0, and increments of 0. 05 for systems with side length ≥ 55σ0. We performed 10 independent trials at each value of εangle. For the largest value of εangle at which at least one of these trials resulted in a complete shell, we then performed 10 additional trials to obtain a more accurate estimate of the shell bending modulus κs at the maximum εangle.
We begin by considering shells with T = 3 spontaneous curvature (Fig 1D). To isolate the effects of cargo on shell size, we consider shell-shell interaction parameters which favor pentamer insertion (setting the ratio of pentamer-hexamer and hexamer-hexamer affinities εPH/εHH ≥ 1. 3) so that assembly without cargo results in primarily T = 3 empty shells for our ratio of pentamer to hexamer concentrations, ρp/ρh = 0. 5, and results in shells close in size to the T = 3 geometry at all of the stoichiometries we consider here. A typical assembly trajectory without cargo is shown in Fig 2A. When simulating assembly around cargo, we set the hexamer-hexamer affinity εHH ≤ 2. 2 (while maintaining εPH/εHH ≥ 1. 3) so that assembly occurs only in the presence of cargo, and we vary cargo-cargo εCC and cargo-shell εSC interaction strengths. Throughout this article, all energy values are given in units of the thermal energy, kBT. Except where mentioned otherwise, values of our simulation shell bending modulus κs fall within the range estimated for β−carboxysomes from AFM nanoindention experiments κs ∈ [1,25]kBT (see Ref. [89] and section ‘Determination of parameter values’ in S2 Text; simulations with shell spontaneous curvature use κs = 10 − 16kBT. We now consider the opposite limit: a system of ‘flat’ hexamer subunits, which have zero spontaneous curvature and thus favor formation of flat sheets (Fig 5A). Fig 5B shows a typical assembly trajectory for flat subunits with εCC = 1. 7, in which the cargo rapidly coalesces followed by adsorption and assembly of the hexamers. Interestingly, the shapes of assembly intermediates reflect the lack hexamer spontaneous curvature—hexamers initially assemble into flat sheet wrapped around the globule, deforming the spherical globule into a cigar shape. Eventually the two sides of the sheet meet, creating a seam with an unfavorable line tension due to unsatisfied subunit contacts. As the seam gradually fills in, the elastic energy associated with such an acute deformation forces the complex toward a more spherical shape. As in systems with spontaneous curvature, the hexamer shells exhibit the 12 five-fold vacancy defects required by topology. If pentamers are present they eventually fill these holes (as in Fig 2 above), but for simplicity we consider systems containing only hexamers here. The large shells are roughly but not perfectly icosahedral, presumably reflecting slow defect reorganization on assembly timescales. The size of the assembled shell is limited by the finite system size of our simulations. Importantly, the same limitation occurs within cells when the cargo undergoes phase separation into a single complex whose size is limited by the enzyme copy number (e. g. the procarboxysome precursor to carboxysome assembly [39,40]). We therefore investigated the dependence of assembly morphologies on system size, as a function of the shell bending modulus, κs (controlled by the parameter εangle). Specifically, at each value of κs we performed a series of simulations in which the maximum size of the cargo globule was controlled by changing the system size with fixed total cargo concentration and hexamer chemical potential (section Materials and methods). An example assembly trajectory for a small system is shown in Fig 5C. As shown in Fig 6, we observe a minimum globule size required for complete shell assembly, which linearly increases with κs. We observe complete wrapping for all system sizes above this threshold. Below the threshold size, assembly stalls with one or more open seams remaining; examples of this configuration are shown for a low and high bending modulus in Fig 6. Interestingly, while the pentameric defects are roughly equally spaced within large shells, small shells assembled with extremely low values of κs tend to exhibit adjacent vacancy pairs (Fig 5C, final frame). This defect morphology focuses curvature in a region with no elastic energy (the vacancy) while reducing the number of unsatisfied hexamer edges. To understand these results, in section S2 Text we present a calculation of the equilibrium shell size distribution for subunits with no spontaneous curvature and stoichiometrically limiting cargo. We restrict the ensemble to spherical shells as observed in the simulations. While the aggregates are large and polydisperse without cargo, the calculation shows that cargo leads to a minimum free energy spherical shell size (S8 and S9 Figs). The linear relationship between minimum shell size and bending modulus can be understood from our equilibrium model by comparing the excess free energy difference ΔΩwrap between the complete shell and an unwrapped globule (see section S2 Text). For the simulated conditions, the size and shape of the cargo globule is essentially the same in each of these states, and thus the free energy difference for a globule wrapped by nh hexamers in Eq. (S2. 9) simplifies to Δ Ω wrap = 8 π κ s + Δ G p + Δ μ h n h (1) with Δμh = ghh + ghc − μh, ΔGp as the free energy due to the 12 pentameric vacancies, ghh (εHH) as the hexamer-hexamer interaction free energy, ghc (εSC) as the hexamer-cargo free energy, and μh = kBT log (ρh) the chemical potential of unassembled hexamers at concentration ρh. The term 8πκs describes the bending energy of the complete shell. The minimum globule size n* corresponds to the locus of parameter values at which ΔΩwrap = 0, giving n * = 8 π - Δ μ h κ s + Δ G p - Δ μ h (2) A linear fit to the simulation results for n* results in Δμh = −2. 4 and ΔGp = 80. 5kBT, or 6. 7kBT per pentameric defect. Plugging in ρh = 10−3 subunits/σ 0 3 and ghc = −8. 1kBT for εSC = 7. 0 (using the estimate from Perlmutter et al. [42]) then results in ghh ≈ −0. 45kBT. This value and the fit value of ΔGp are reasonably close to interactions estimated from the relationship between the shell-shell dimerization free energy ghh and potential well-depth εHH for a similar model in Perlmutter et al. [42]. Thus, the simulation results are consistent with the minimum stable shell size predicted by the theory. We have used computational and theoretical modeling to investigate factors that control the assembly of a protein shell around a fluid cargo. We have focused on two limiting regimes of protein interaction geometries—high spontaneous curvature that drives the formation of small shells, and zero spontaneous curvature that favors assembly of flat sheets or polydisperse shells. In both regimes the presence of cargo can significantly alter the size distribution of assembled shells. For high spontaneous curvature, encapsulated cargo tends to increase shell size, whereas for shell proteins with low (or zero) spontaneous curvature cargo templating provides a mechanism to drive shell curvature and thus tends to reduce shell size. These results could provide a qualitative explanation for experimental observations on different systems in which full microcompartment shells were either larger or smaller than empty shells [23,28,31,37,38]. Our simulations identify a combination of kinetic and thermodynamic mechanisms governing microcompartment size control. At equilibrium, the shell size is determined by the stoichiometry between cargo and shell subunits, with an excess of cargo or shell protein respectively favoring larger or smaller shells. Similarly, a high surface energy (high cargo surface tension and weak shell-cargo interactions) favors larger shells whereas a strong shell bending modulus favors shells closer to the preferred size. Although dynamical simulations exhibit similar qualitative trends to these equilibrium results, we observe significant kinetic effects as well. Fast cargo coalescence relative to rates of shell assembly favors larger shells, since closure of an assembling shell prevents further cargo aggregation. Thus, the shell size is strongly correlated to the assembly pathway, with two-step assembly leading to larger shells than single-step pathways. Although many factors likely control shell size in biological systems, this result is consistent with the observations of small empty shell assemblies [23,28,31,37] and the fact that β-carboxysomes (which assemble by two step pathways [39,40]) tend to be larger and more polydisperse than α-carboxysomes (which experiments suggest assemble by one-step pathways [33,41]). Our results for shell proteins without spontaneous curvature build upon Rotskoff and Geissler [58], which identified a kinetic mechanism in which cargo templating drives shell curvature, and shell closure eventually arrests assembly. Their mechanism proceeds by two-step assembly, with initial nucleation of a cargo globule followed by assembly of shell subunits, but requires that rates of subunit arrival are at least 10 times faster than cargo arrival rates [58]. However, it is unclear how many physical microcompartment systems may fit this criteria, and our results suggest other mechanisms may play important roles in microcompartment assembly. Firstly, if cargo is stoichiometrically limiting then the finite-pool mechanism can result in finite shell sizes, with the coalesced cargo still providing a template for shell curvature. Secondly, subunits with spontaneous curvature can form complete shells even under conditions of excess cargo or fast coalescence rates that lead to large cargo aggregates (Fig 3D), as observed for carboxysome assembly in cells [39]. Thus, biological microcompartments with some degree of preferred shell curvature could robustly assemble over a much wider parameter space than systems without spontaneous curvature. Intriguingly, the recent atomic-resolution microcompartment structure from Sutter et al. [31] suggests that different hexamer or pseudo-hexamer species have different preferred subunit-subunit angles, and thus the spontaneous curvature may depend on the shell composition. We will investigate this in a future work. The importance of spontaneous curvature to a particular BMC system could be investigated by comparing our computational predictions to experimental shell size distributions measured for varying cargo/shell protein stoichiometries and interaction strengths. While such tests would be most straightforward to perform in vitro, they could be performed in vivo by varying expression levels of various shell proteins or the enzymatic cargoes. Of particular interest would be a comparison between the shell size distribution in the presence and absence of cargo. However, note that we have focused on extreme limits (high spontaneous curvature or zero spontaneous curvature); systems with moderate shell spontaneous curvature may exhibit less dramatic cargo effects. Also note that the effective shell spontaneous curvature depends on the stoichiometries of different shell protein species; e. g. , overexpressing pentamers would shift the size distribution toward smaller shells (Fig 2D). These results have implications for targeting new core enzymes to BMC interiors. Recent experiments have shown that alternative cargoes can be targeted to BMC interiors by incorporating encapsulation peptides that mediate cargo-shell interactions, but that relatively small amounts of cargo were packaged [21–23,96]. Our previous simulations showed that assembly of full shells requires both cargo-shell and cargo-cargo (direct or mediated) interactions. Here, we see that the strength of cargo-cargo interactions can not only affect the efficiency of cargo loading, but also the size of the containing shell.
Bacterial microcompartments are protein shells that encase enzymes and reactants to enable bacteria to perform vital reactions, such as breaking down chemicals for energy or converting the products of photosynthesis into sugars. Microcompartments are essential for many bacteria, including human pathogens. Thus, there is great interest in understanding how microcompartment shells assemble around their cargo (the interior enzymes and reactants), and what determines the structure and size of a microcompartment. These questions are difficult to answer with experiments alone, because most intermediates in the assembly process are too short-lived to characterize in experiments. Therefore, this article describes theoretical and computational models for microcompartments, which predict assembly pathways and how the sizes of assembled shells depend on factors such as protein interactions and concentrations. The simulations show that the properties of the cargo are an important factor for determining shell size, and suggest an explanation for recent experimental results showing that cargo can either increase or decrease shell size. In addition to helping to understand the natural behavior of microcompartments, the simulations provide guidance to researchers working to reengineer microcompartments to produce drugs or biofuels.
protein interactions
condensed matter physics
geometry
cargo proteins
plant science
mathematics
photosynthesis
ribulose-1,5-bisphosphate carboxylase oxygenase
thermodynamics
proteins
chemistry
physics
biochemistry
stoichiometry
plant biochemistry
curvature
biology and life sciences
physical sciences
nucleation
symmetry
The role of the encapsulated cargo in microcompartment assembly