Equilibrium mechanisms of self-limiting assembly

Self-assembly is a ubiquitous process in synthetic and biological systems, broadly defined as the spontaneous organization of multiple subunits (macromolecules, particles, etc.) into ordered multiunit structures. The vast majority of equilibrium assembly processes give rise to two states: one consisting of dispersed disassociated subunits and the other consisting of a bulk-condensed state of unlimited size. This review focuses on the more specialized class of self-limiting assembly, which describes equilibrium assembly processes resulting in finite-size structures. These systems pose a generic and basic question, how do thermodynamic processes involving noncovalent interactions between identical subunits “measure” and select the size of assembled structures? This review begins with an introduction to the basic statistical mechanical framework for assembly thermodynamics that is used to highlight the key physical ingredients ensuring that equilibrium assembly will terminate at finite dimensions. Then examples of self-limiting assembly systems are introduced, and they are classified within this framework based on two broad categories: self-closing assemblies and open-boundary assemblies. These include well-known cases in biology and synthetic soft matter (micellization of amphiphiles and shell and tubule formation of tapered subunits) as well as less widely known classes of assemblies, such as short-range attractive or long-range repulsive systems and geometrically frustrated assemblies. For each of these self-limiting mechanisms, the physical mechanisms that select equilibrium assembly size, as well as the potential limitations of finite-size selection, are described. Finally, alternative mechanisms for finite-size assemblies are discussed, and contrasts are drawn with the size control that these can achieve relative to self-limitation in equilibrium, single-species assemblies.

Figure 1

Functional, finite-sized assemblies of proteins in biology: (a) protein shells of clathrin (left panel) and viral capsids of Herpes simplex (right panel), (b) photonic nanostructures formed by keratin aggregates in feather barbs of plum-throated continga (inset), and (c) finite-diameter fibers in reconstituted fibrin clot. (a), left panel: adapted from [239]. (a), right panel: adapted from [8]. (b): adapted from [64]. (c): adapted from [304].

Figure 2

Schematic illustrations of two classes of SLA described in Sec. 3. (a) Self-closing assembly, in which inter-subunit rotations lead to cohesive assembly into closed, boundary-free aggregations. (b) Open-boundary (self-limiting) assembly, in which intra-aggregate stress accumulates with assembly and restrains the cohesive drive toward unlimited size.

Figure 3

(a) Examples of $d$-dimensional (linear, planar, and spherical) short-range cohesive aggregation. (b) Plots of the aggregation distributions (relative counts of $n$-mers) for $d=1$ (top panel) and $d=2$ (bottom panel) for monomer concentrations increasing to saturation (i.e., ${\varphi }_1={\varphi }_{\mathrm{s}}$) for ${\mathrm{\Delta }}_0=1k_{B}T$. While the dispersity (and mean size) of linear aggregates diverges as ${\varphi }_1\to {\varphi }_{\mathrm{s}}$, it remains finite for $d\ge 2$ at saturation. (b) Equation of state (ideal aggregation theory) for the free monomer population ${\varphi }_1$ as a function of the total concentration $\mathrm{\Phi }$ for linear, planar, and spherical aggregates for ${\mathrm{\Delta }}_0=2.75k_{B}T$. For $d=2$ and $d=3$ the free monomer concentration saturates at a finite $\mathrm{\Phi }$ where ${\varphi }_1={\varphi }_{\text{s}}$. For linear aggregation, saturation is not reached in the ideal theory.

Figure 4

(a) Schematic plot of the aggregation free energy per subunit shown as a continuous function of aggregation number $n$. The dashed line shows a harmonic expansion around a local minimum at the target size $n=n_{\mathrm{T}}$. (b) Schematic plots of the aggregate distribution for a model of the type sketched in (a).

Figure 5

(a) Plots of a “two-state” model composed only of monomers ($n=1$) and aggregates of a single peak size ($n=n_{*}\approx n_{\mathrm{T}}$) as functions of the total concentration $\mathrm{\Phi }$ and for different finite aggregate sizes. The critical aggregation concentration (CAC), here ${\varphi }_{*}$, characterizes the concentration range beyond which aggregates dominate the subunit population. (b) Assembly behavior of hepatitis B virus (HBV) capsid protein in vitro as a function of concentration. The concentration dependence of the fraction of subunits (protein dimer) is shown in two states: free subunits and assembled capsids composed of 120 subunits. Notice that as the total concentration crosses the CAC (labeled $K_{D,\mathrm{app}}$) the concentration of free subunits is nearly constant, with almost all additional subunits assembling into capsids. From [240].

Figure 6

(a) Schematic plot of the aggregation energetics for a case with two local minima corresponding to small and large finite aggregates containing $n_{\mathrm{S}}$ and $n_{\mathrm{L}}$ respective subunits in their target sizes. The solid curve shows a case where small aggregates are the global minimum of $\epsilon (n)$, while the dashed curve shows a case where the larger aggregate is the global minimum. The latter case can lead to secondary CAC transitions between small and large aggregates, as shown in (b), which plots the monomer, small aggregate, and large aggregate populations as functions of total concentration for large aggregates that are slightly energetically more favorable than small ones ($e^{\beta ({\epsilon }_{\mathrm{L}}-{\epsilon }_{\mathrm{S}})}=0.8$). In this case, small aggregates dominate at intermediate concentrations but ultimately are overtaken by a second population of large aggregates above a second CAC. (c) Assembly state map for the two finite aggregate model, as a function of the per subunit energy difference between small and large aggregates and total concentration. The color scale shows the mean aggregate size, while the solid lines indicate boundaries between states dominated by monomers, $n_{\mathrm{S}}$-mers, and $n_{\mathrm{L}}$-mers. The dashed lines indicate where subdominant aggregates reach the monomer concentration. (b),(c) Results of the two finite aggregate model for $n_{\mathrm{S}}=10$ and $n_{\mathrm{L}}=50$.

Figure 7

(a) Schematic plot of polymorphic aggregation energetics with three competing branches of assembly: finite aggregates with a local minimum at $n_{\mathrm{F}}$, 1D aggregates, and bulk aggregates ($n\to \infty$ energy shown as a horizontal dashed line). In this case, the infinite 1D aggregate has a lower per subunit energy than finite aggregates, and there is a barrier in total energy $\delta$ that separates these states at $n=n_{\mathrm{F}}$, i.e., the double arrow in (a) corresponds to $\delta /n_{\mathrm{F}}$. (b) Phase diagram for concentration-dependent size selection. The dominant aggregation state is shown for a system with coexistence among finite aggregates with $n_{\mathrm{F}}=100$ subunits, separated by an energy gap ${\epsilon }_{\mathrm{F}}-{\epsilon }_{1\mathrm{D}}$ and a barrier of $\delta$ [Eq. (34)] to 1D aggregates. There is an additional per subunit energy gap of ${\epsilon }_{1\mathrm{D}}-{\epsilon }_{\mathrm{bulk}}=0.0005k_{B}T$ between the 1D and bulk aggregates. The horizontal axis gives the energy gap between spheres and cylinders, and the vertical axis gives the total concentration relative to the CAC for finite aggregates ${\varphi }_{\mathrm{F}}^{*}\simeq e^{\beta {\epsilon }_{\mathrm{F}}}/n_{\mathrm{F}}$. The boundaries between monomers, finite aggregates, and 1D aggregates are determined by crossovers in the most populous aggregate type from Eq. (30), while the point of bulk saturation is determined by the point when $\mu ={\epsilon }_{\mathrm{bulk}}$. In this example, the energy per subunit in finite aggregates is fixed at ${\epsilon }_{\mathrm{F}}=-10k_{B}T$ and the end cap energy of the spherocylinders is ${\mathrm{\Delta }}_0=20k_{B}T$. The maximum energy gap for which second CAC behavior occurs [$\mathrm{\Delta }{\epsilon }_{\max }\approx 0.14k_{B}T$; Eq. (35)] is indicated along the $x$ axis.

Figure 8

(a) Schematic of spherical shell assembly with two independent self-closing directions of assembly. (b) Schematic of tubule assembly with one self-closing direction (circumferential) and one unlimited direction (axial) of assembly.

Figure 9

(a) Schematic geometry of a “fluid capsule” of tapered subunits that assumes partial shell geometries of spherical caps with aperture angle $0<\mathrm{\Theta }\le \pi$. (b) Contour map of the energy density landscape of the capsule model as a function of aperture angle and the ratio of subunits $n$ to the preferred number in the ideal closed shell $n_{\mathrm{T}}$. Low (high) values of $\epsilon (n)$ appear in purple (red). The dotted line shows a partial shell with the target curvature, while the solid line indicates the minimal-energy cap, whose curvature radius is slightly compressed by the line tension of the boundary. Beyond a threshold cap size $n=n_{\mathrm{S}}\simeq 0.4n_{\mathrm{T}}$, this open cap becomes unstable to preclosure and the minimal-energy branch runs along $\mathrm{\Theta }\to \pi$. This landscape corresponds to a dimensionless line tension $\overline{\lambda }=0.1$. (c) Plots of the minimal energy branches of assembly. Incomplete caps are shown as colored curves (metastable portions are dashed), and the closed shell ($\mathrm{\Theta }\to \pi$) is shown as a black curve. Inset: the size $n_{\mathrm{S}}$ corresponding to the preclosure, or “snap,” transition between stable open caps and closed shells as a function of line tension.

Figure 10

Examples of well-formed and defective woodchuck hepatitis B virus (WHV) capsids. (a) Portion of a representative cryo-EM image of empty WHV capsids assembled in vitro in the absence of RNA (from capsid proteins with the C-terminal RNA binding deleted, wCp149). The population of capsids is structurally heterogeneous; the black arrow indicates an example of a $T=4$ icosahedral capsid, while the white arrows indicate examples of defective capsids. (b) 2D class averages of (1) icosahedral and (2)–(4) defective particles. The red arrows in (2) and (4) indicate locations where the capsid shell is overgrown and overlaps itself. (c) Schematic model of a $T=4$ icosahedral HBV capsid, with the monomers forming the 12 fivefold vertices colored green, and the others colored blue. (d) Hypothetical model of the structure of a capsid that is nonicosahedral but closed, with an elongated structure containing 150 protein dimers (the icosahedral capsid has 120 dimers). Here the green dimers are in pentamers or extend between pentamers and hexamers, and the blue dimers are in hexamers or extend between two hexamers. Adapted from [223].

Figure 11

(a) Sketch of a single-tailed surfactant molecule. (b) Schematic of a spherical micelle with the thickness of the solvophobic core $\ell$ and the area per head group $a$ highlighted. A wedgelike portion of the micelle is cut away to illustrate the interior packing of tails. (c) Spherocylinder model of a wormlike micelle, which has a cylindrical portion of length $L$ capped by two hemispherical micelles of equal radius. (d) Energy landscape for spherocylinders described by the model in Eq. (44) with $P^{-1}=2.25$ and $\overline{k}=0.1$. The dashed orange line shows the $L=0$ spherical micelle branch and the dashed pink line shows the $L=\infty$ wormlike micelle branch, whose corresponding energy as a function of reduced aggregation number is plotted in (e).

Figure 12

(a) Assembly state phase diagram for the amphiphilic aggregate model in Eq. (44) for dimensionless stiffness $\overline{k}=1$ and $v_0^2/a_0^3=10$, where ${\varphi }_{*}$ is the nominal CMC for cylinders. Solid lines mark the boundaries between the most populous aggregate type, and dashed lines indicate the boundaries in the infinite $\mathrm{\Phi }$ limit. Inset: enlargement near the boundary between cylinders and bilayers illustrating an extremely narrow window of secondary CMC behavior due to the large mean (finite) size of cylinders. (b) Summary of the polymorphic assembly of the amphiphile model in the plane of stiffness $\overline{k}$ and inverse packing parameter $P^{-1}$ for $v_0^2/a_0^3=10$. Regimes of single CMC behavior are shown in solid red, white, and blue for bilayers, cylinders, and spheres, respectively. The regime of polymorphic concentration-driven sphere-to-spherocylinder transitions is colored on a blue-green scale according to the ratio of the second CMC (spheres to spherocylinders) to the first CMC (monomers to spheres). (c)–(f) Cryo-transmission electron micrographs of micelles formed by dimeric (gemini) surfactancts, at $25\hspace{0.17em}°\mathrm{C}$ at increasing weight percent: (c) 0.26%, (d) 0.5%, (e) 0.62%, and (f) 0.74% (scale bars equal 100 nm). The coexistence of cylindrical micelles of increasing total length for (d)–(f), and absence of lengths intermediate to spheres and the shortest cylinders is consistent with a concentration-dependent second CMC. The inset of (f) shows the bulbous ends of a cylindrical micelle (scale bar, 25 nm), which is consistent with an energy barrier between spherical and long cylinders due to a mismatch in preferred radius. (c)–(f) Adapted from [19].

Figure 13

(a) Schematic plot of the aggregation energetics per subunit for self-limiting open-boundary assembly, where the blue, red, and black curves, respectively, show the surface energy density, excess energy, and total energy density. The dashed curve shows power-law growth of excess energy at small size. The red point highlights the equilibrium self-limiting width $W_{*}$. Inset: variation of this equilibrium width with increasing surface energy $\mathrm{\Sigma }$ assuming a fixed ${\epsilon }_{\mathrm{ex}}(W)$. (b) Schematic plot of the equilibrium width as a function of $\mathrm{\Sigma }$, where the dashed portion indicates the possibility of a finite-width branch that is metastable relative to the bulk state $W\to \infty$. The boundary between equilibrium self-limiting and bulk states is marked by a maximum self-limiting size $W_{\max }$, denoted as the escape size.

Figure 14

(a) Schematic plots of excess energy vs width for two different models of OBA: model 1 (blue curve) and model 2 (orange curve). Both cross over from power-law growth at small size to saturating excess energy at large size but do so with different functional forms (the asymptotic approach to ${\epsilon }_{\infty }$ is slower in model 2 than in model 1). The area of the blue rectangles graphically illustrates the definition of the accumulant in Eq. (50). (b) Accumulant $\mathcal{A}(W)$ as a function of width for the two models shown in (a). Regions of increasing $\mathcal{A}(W)$ correspond to possible ranges of equilibrium self-limiting sizes for a given form of ${\epsilon }_{\mathrm{ex}}(W)$. That is, values of $W$ for which ${\mathcal{A}}^{\prime }(W)>0$ correspond to energy minima of $\epsilon (W)$, which are lower in energy than the bulk state $\epsilon (W\to \infty )$, for a particular value of the surface energy $\mathrm{\Sigma }$ given by Eq. (47). Hence, model 1 shows an upper limit for maximal self-limiting size $W_{\max }$, while model 2 exhibits stable self-limiting equilibria at all sizes.

Figure 15

(a) Schematics of aggregates of particles interacting via short-range (i.e., contact) interactions and long-range repulsions, with a screening length ${\kappa }^{-1}$, which can be large or small relative to the aggregate size $R$. (b) Excess energy of spherical aggregates vs radius showing a crossover from power-law growth for $R\ll {\kappa }^{-1}$ to an asymptotically saturating bulk energy for $R\gg {\kappa }^{-1}$. (c) Plot of the equilibrium finite radius $R_{*}$ of the SALR model vs surface energy $\mathrm{\Sigma }$ showing a continuous divergence to the bulk at a finite $\mathrm{\Sigma }={\mathrm{\Sigma }}_{\max }$. The second-order-like transition from finite to bulk states predicted by this model (solid curve) is consistent with the monotonically increasing form of the accumulant plotted in the inset, where this result assumes that the bulk state has uniform density. Assuming instead that the bulk state has a lower-energy, nonuniform density [such as a periodic aggregate morphology ([328])], the transition will be first order, as illustrated by the nonmonotonic accumulant shown as a dashed line. (d) Confocal microscope image of clusters of charged colloidal particles (radius 660 nm). In these experiments short-range attractions are induced by depletion forces generated by inert polymers in suspension, while electrostatic repulsions are maintained at relative long range due to the low concentration of mobile ions in the host organic solvent. Adapted from [274].

Figure 16

(a) Heuristic warped jigsaw model for GFA from [92], in which directional interactions promote curvature along rows of a locally preferred 2D “lattice.” Assembly in both bonding directions leads to “misfits,” as in the tetrameric aggregate. (b) Schematic plot of the excess energy per subunit for the warped jigsaw model due to the superextensive buildup of elastic costs of misfits. Intra-assembly strains are illustrated via particle color, from unstrained (yellow) to highly strained (red). The excess energy shows the characteristic crossover from power-law growth at small aggregates to an asymptotic approach to a strained bulk state (in this case envisioned as shape flattening of jigsaw particles). (c) Schematic phase diagram for a generic model GFA in Eq. (59), considered (at fixed concentration and temperature) as a function of the ratio of surface energy to elastic stiffness and a measure of the strength of frustration $f_0$.

Figure 17

(a) Schematic of a chiral, crystalline ribbon, where colors indicate the local extensional strains required by negative Gaussian curvature, from low (blue) to high (red). (b) Plots of the equilibrium shape relaxation of the narrow-ribbon model of [82] as a function of increasing ribbon width for curvature components $C_{ij}$ and Gaussian curvature $K_{\mathrm{G}}$, which is approximately uniform over the ribbon. The solid branches show the minimal-energy configurations, while the dashed line indicates the unstable helicoidal equilibrium. Widths are scaled by the characteristic length $W_{\mathrm{un}}$ defined in Eq. (63). (c) Schematics of the shape equilibrium, in particular, the shape transition from helicoids to spiral ribbons at a critical width $W_{\mathrm{c}}$. (d),(e) Respective excess energies for helicoidal (dashed lines) and helicoid-spiral shape branches (solid lines). While both shape modes predict an asymptotically relaxed frustration energy and a finite self-limiting width, spiral ribbons expel $K_{\mathrm{G}}$ at a much faster rate, leading to a narrower range of self-limitation. (e)–(g) Transmission electron microscopy images of ribbons assembled from chiral bola-amphiphiles. Illustrated are morphologies evolving with assembly time (scale bars, 100 nm): in (e) helicoids are observed after 24 hours, in (f) spiral ribbons are observed after one week, and, finally, in (g) closed tubules are observed after 5 months. Adapted from [328].

Figure 18

The grand free energy $\mathrm{\Omega }$ for the capsid model with fixed curvature $R_{\mathrm{T}}$ [Eq. (70)] as a function of partial capsid size $n$. (a) $\mathrm{\Omega }(n)$ for indicated complete capsid sizes $n_{\mathrm{T}}$, with the limit $n_{\mathrm{T}}\to \infty$ corresponding to unlimited assembly (a flat disk). The calculation is performed for a chemical potential difference $\mathrm{\Delta }\mu =-{\epsilon }_{\mathrm{T}}-\mu =-4k_{B}T$ and a per subunit binding free energy ${\epsilon }_{\mathrm{T}}=-15k_{B}T$, corresponding to a subunit-subunit contact energy of $7.5k_{B}T$ and tetravalent subunits ([336]; [42]). We set the line energy to $\lambda ={\epsilon }_{\mathrm{T}}/2a_0^{1/2}$, corresponding to one unsatisfied contact per subunit on the partial shell rim. (b) $\mathrm{\Omega }(n)$ as a function of the chemical potential difference $\mathrm{\Delta }\mu$ for ${\epsilon }_{\mathrm{T}}=-15k_{B}T$ and complete capsid size $n_{\mathrm{T}}=120$. In (a) and (b), the solid lines correspond to sizes for which the quasiequilibrium approximation described in the text applies (i.e., $n\le n_{\mathrm{nuc}}$), and thus the concentration of intermediates is given by $\varphi (n)\propto e^{-\beta \mathrm{\Omega }(n)}$. The dashed lines correspond to sizes $n>n_{\mathrm{nuc}}$ for which this assumption is not valid and the intermediate concentrations cannot be described by a quasiequilibrium (except approximately for the case $\mathrm{\Delta }\mu =0$).

Figure 19

(a) Assembly “phase diagram” for the capsid model ($B\to \infty$ model of the fluid capsule model). The boundaries between the different kinetics regimes discussed in the text are shown as a function of capsid size and supersaturation $\mathrm{\Phi }/{\mathrm{\Phi }}_{\mathrm{s}}=e^{-\beta \mathrm{\Delta }\mu }$. The calculation was performed using Eqs. (68), (5), and (71) with $\alpha =1$, as well as Eq. (c2). We set ${\epsilon }_{\min }=-15k_{B}T$ and $\lambda ={\epsilon }_{\mathrm{T}}/2a_0^{1/2}$ as in Fig. 18. (b),(c) TEM images of in vitro assembly of empty capsids from cowpea chlorotic mottle virus capsid proteins. (b) corresponds to productive assembly, while (c) corresponds to assembly of long-lived partial shells (the monomer starvation trap) that occurs under stronger subunit-subunit interactions. From [336].