Equilibrium mechanisms of self-limiting assembly

Ideal aggregation theory is well established for certain classes of self-assembly systems, particularly in the context of amphiphiles and surfactants. As such, this theory is better described, and in greater depth, in references such as Tanford (1974); Israelachvili et al. (1976); Safran (1994); Gelbart et al. (1994). Here, our purpose is to consider the application and implications of ideal aggregation theory to a broader class of self-limiting assembly systems. Hence, we only present a minimal introduction to the elements necessary to describe aggregation to self-limiting states.

We consider a solution of $N$ total subunits in a fixed total volume $V$. In what follows, we refer to unnassembled single subunits as monomers ¹¹1In literature on amphiphile aggregation the term “unimer” is often used to describe the single subunit, to avoid overlap with the connotation of “monomer” as the chemical repeat of macromolecular chain, which is often a component of self-assembling molecular subunits.. To describe the concentration of subunits, it is convenient to scale concentration by the reference state concentration $v_{0}^{-1}$; i.e., $v_{0}$ is the volume per subunit in the reference state ²²2For concreteness, we may take $v_{0}$ to be the subunit volume in the disassociated state, which provides a convenient and non-dimensional measure of concentration which will be much less than unity for dilute conditions. However, the formalism is independent of choice of reference state; for example, $v_{0}=1/N_{\mathrm{AV}}$ liter with the reference state concentration of 1 mol/liter commonly used in the life sciences. Changing the definition of the reference state only uniformly rescales the free energy of intra-aggregate interactions $\epsilon(n)$.. In this way, we can define concentration in non-dimensional terms as the total volume fraction of subunits, $\Phi=Nv_{0}/V$ ³³3Throughout, we also refer to a $\Phi$ more colloquially as the “concentration” of subunits. The $N$ subunits are distributed among distinct $n$-mer aggregates, with the volume fraction of subunits in $n$-mers defined as $\phi_{n}$. In the following, we refer to $\phi_{n}$ as the subunit fraction distribution. Throughout this review, it is also useful to introduce a separate variable for the aggregate distribution $\rho_{n}\equiv\phi_{n}/n$, which describes the relative count of $n$-mers in the mixture. Defined in this way, $\phi_{n}$ and $\rho_{n}$ are all less than unity for all $n$. Moreover, for the particular assumptions of ideal aggregation to hold (i.e. two-body contacts between aggregates are vanishingly rare), these quantities must all remain much less than unity. To simplify the nomenclature, throughout the review we refer to the non-dimensional concentration (volume-fraction) simply as the concentration.

We define $n\epsilon(n)$ as the free energy of intra-aggregate interactions; i.e., $\epsilon(n)$ is the per subunit aggregation free energy in an $n$-mer. The total free energy $F$ for the ideal distribution of aggregates is given by

with the two terms in the parentheses respectively representing the intra-aggregate interaction free energy and translational entropy (in the ideal solution approximation) of $n$-mers, with the $1/n$ in the latter term reflecting the critical fact that all subunits of an $n$-mer share a common, single center-of-mass degree of freedom.

To obtain the equilibrium aggregate size distribution, we minimize $F$ with respect to $\phi_{n}$, subject to the constraint that the total subunit concentration is fixed:

i.e.,

with $\mu$ playing the role of a Lagrange multiplier. This yields

showing that $\mu$ is the subunit chemical potential. This condition requires that subunits have the same chemical potential in all aggregates, and it derives from both the energetics of the assembly and the (ideal) translational entropy of the $n$-mer. Note that the prefactor of $1/n$ of the translational entropy, deriving from the sharing of a single center of mass in an $n$-mer, reflects a generically higher translational entropy of disaggregated states. The limit $n\to\infty$ gives $\mu=\epsilon(\infty)$, describing the equilibrium between monomers and a bulk phase-separated condensate which has no translational entropy. For simplicity of notation, throughout this article, we choose to define energies such that $\epsilon(1)=0$, in which case $\epsilon(n)$ is defined as the difference of the per subunit energy between an $n$-mer and disassembled monomers.

It is convenient to use the first equality in eq. (4) to recast the chemical potential in terms of the (unknown) monomer concentration, from which we can reformulate the generic chemical equilibrium conditions in terms of the law of mass action

where $\beta^{-1}\equiv k_{\text{B}}T$. Inserting the expression for $\phi_{n}(\phi_{1})$ into the fixed number concentration eq. (2) and summing over $n$ then results in an equation of state relating the total concentration $\Phi$ to the monomer concentration $\phi_{1}$. This equation of state and the underlying distribution of assemblies, $\phi_{n}$, derive from the specific $n$-dependence of aggregate interactions, with equilibrium states that are dominated by self-limited aggregates occurring only for certain forms of $\epsilon(n)$.

Before describing models that give rise to SLA, we first consider the thermodynamics of the simplest models of physical association, described by short-range attraction between subunits. Crucially, while these models are relevant to a broad range of physical scenarios like colloidal crystallization Manoharan (2015); Morphew et al. (2018), they do not exhibit SLA. Yet, they will serve as a useful reference point for illuminating the necessary conditions for SLA. Specifically, these models result in either a single dispersed state whose most populous aggregate state is $n=1$ (the free monomer), or coexistence between the dispersed monomer-dominated state and an unlimited aggregate (macrophase separation). The absence of equilibrium finite-sized states will be traced to the generic size-dependence of the cost of open boundaries at the edges of cohesive aggregates. In subsequent sections we will show mechanisms that compete against the open boundary cost to enable SLA.

Here, we consider models where inter-subunit association promotes uniform $d$-dimensional aggregates, e.g. 1-dimensional chain-like or 2-dimension sheet-like aggregates. Every internal subunit forms, on average, $z$ attractive bonds of strength $-u_{0}$, and subunits at the free boundary have $\delta z$ fewer contacts (e.g. Fig. 3A). For example, in $d=1$ chain-like assembly $z=2$ and $\delta z=1$. For the generic dimensionality, the interaction free energy takes the form

where $\epsilon_{0}=u_{0}z/2$ and represents the per subunit cohesive free energy in the bulk (i.e. $n\to\infty$) structure. The second term derives from the growth of number of particles at the boundary ($\sim n^{(d-1)/d}$) and their deficit of cohesive bonds ($\delta z$), so that $\Delta_{0}$ is equal to $u_{0}(\delta z)$ times a geometric factor accounting for the mean bond geometry at the boundary. Notably, the bonding-geometry in these assemblies permits the structure to grow uniformly without disrupting this local contact structure at any size scale, a condition that we revisit when describing examples of self-limiting assembly in Sec. III.

We define the concentration $\phi_{\rm s}\equiv e^{-\beta\epsilon_{0}}$, so that the law of mass action, eq. (5), takes the form

where $\alpha=1-1/d$ is an exponent that characterizes the geometric growth of the exposed boundary with $n$. As shown in Fig. 3B, $\phi_{1}\leq\phi_{\rm s}$ and the distribution $\rho_{n}=\phi_{n}/n$ decreases exponentially with aggregate size for large $n$, and for any $d$. Hence, under these conditions $\Phi$, the sum over the subunit fraction distribution in eq. (2), is finite, implying the existence of conditions where the concentration of subunits achieves equilibrium in the suspension. However, no such equilibrium exists for $\phi_{1}>\phi_{\rm s}$, implying that $\phi_{1}\to\phi_{\rm s}$ is an upper limit to concentrations that may be in equilibrium in a dispersed state. In other words, when $\Phi$ is sufficiently large that $\phi_{1}=\phi_{\rm s}$ the solution is saturated, and additional subunits (further increasing $\Phi$) must phase separate to the macroscopic state (i.e. $n\to\infty$).

First consider the linear case ($d=1$), where the equation of state can be ready computed from eq. (7) with $\alpha=0$ and the geometric series,

which notably diverges as $\phi_{1}\to\phi_{\rm s}$. As plotted in Fig. Fig. 3C, this divergence indicates that the monomer concentration increases with total concentration, but never reaches the point of saturation (i.e. $\phi_{1}<\phi_{\rm s}$ for any finite $\Phi$). Hence, for all subunit concentrations the system maintains $\phi_{1}/\phi_{\rm s}<1$, implying that the distribution of linear aggregates is always exponential, $\phi_{n}/n\propto e^{-n/\langle n\rangle}$, where the number-average length is $\langle n\rangle=1/\ln(\phi_{\rm s}/\phi_{1})$. ⁴⁴4The number-average aggregate size is $\langle n\rangle=(\sum_{n}n\rho_{n})/(\sum_{n}\rho_{n})=(\sum_{n}\phi_{n})/(\sum_{n}\phi_{n}/n)$, where $\phi_{n}/n$ is the distribution of $n$-mers, while the mass-average aggregate size is $\langle n\rangle_{\text{M}}=(\sum_{n}n\phi_{n})/(\sum_{n}\phi_{n})$. Noting that $\epsilon_{0}=\Delta_{0}$ for a 1D chain assembly, the growth of mean length with end energy in the limit of high concentration $\langle n\rangle\simeq e^{\beta\Delta_{0}/2}\sqrt{\Phi}$, which is well-known for equilibrium polymers  Hiemenz and Lodge (2007) and cylindrical micelles Safran (1994); Gelbart et al. (1994), highlights the mechanism that prevents “bulk” assembly for 1D aggregation. In this dimension, the probability to introduce a free end remains finite $\sim e^{\beta\Delta_{0}}$ in the $n\to\infty$ limit, analogous to the statistics of domain walls in the 1D Ising model at finite temperature Fisher (1984). However, while the mean-size is finite, this case is distinct from what we will describe as self-limiting assembly, in both the strong dependence of $\langle n\rangle$ on total concentration, and perhaps more significantly, the fact that fluctuations in aggregate size are always comparable to the mean; that is, $\big{\langle}(n-\langle n\rangle)^{2}\big{\rangle}^{1/2}\propto\langle n\rangle$.

For higher assembly dimensionality, ($d>1$), the geometric growth of the free boundary cost restrains the $n\to\infty$ divergence in the distribution eq. (7) as the solution approaches saturation. For $\phi_{1}=\phi_{\rm s}$ the distribution takes the form $\phi_{n}(\phi_{1}\to\phi_{\rm s})=ne^{-\beta\Delta_{0}n^{\alpha}}$, the sum over which converges for $\alpha>0$ when $d>1$. For example, we can approximate the sum for planar aggregates ($d=2$) by replacing the sum over aggregation number $n$ with dimensionless aggregate radius $n=\pi r^{2}$, i.e. $\sum_{n}\phi_{n}\to 2\pi\int drr~\phi(r)$. At saturation, aggregates sizes are exponentially distributed, $\phi(r)\simeq\pi r^{2}e^{-\sqrt{\pi}\beta\Delta_{0}r}$, yielding a saturation concentration

at which point the ideal solution of aggregates reaches equilibrium with the bulk condensate. Thus, with the exception of the special case of $d=1$ assembly, short-range cohesive aggregation is characterized by a finite saturation concentration, $\Phi_{\rm s}=\Phi(\phi_{1}=\phi_{\rm s})$, above which subunits phase separate into an unlimited bulk structure.

The thermodynamics of these examples are plotted in Fig. 3C in terms of the equation of state $\phi_{1}(\Phi)$ for each dimensionality. Note that while the distributions of short-range interacting systems have finite mean sizes in the absence of macrophase separation, their distributions are dominated by monomers; i.e., the concentration of $n$-mers $\rho_{n}$ is always maximal for $n=1$, a property that sharply contrasts with the SLA behavior described next in Sec. II.2.

We first describe the simplest picture of the concentration dependence of ideal aggregation. We begin with the assumption of an energy per subunit $\epsilon(n)$ of the form shown in Fig. 4A, which has a single energy minimum at a finite aggregation number $n_{\rm T}$, which we call the target size. The basic dependence of aggregation on concentration for such a model is sketched in Fig. 4B. There are two dominant populations of aggregates, monomers and $n$-mers, with the $n$-mers narrowly distributed around the most populous state $n_{*}\approx n_{\rm T}$.

For large enough $n_{*}$, the thermodynamics of aggregation can be captured, to a first approximation, by a two-state, or bimodal, distribution, in which fluctuations around free monomers and the $n$-mer aggregate peak are neglected. We will self-consistently test the validity of this approximation below (see eq. 18). When subunits are distributed strictly between the $n=1$ and $n=n_{*}$ states, the conservation of subunit mass is simply $\Phi=\phi_{1}+\phi_{n_{*}}$. Chemical equilibrium then gives the concentration in preferred aggregates $\phi_{n_{*}}=n_{*}\big{(}\phi_{1}e^{-\beta\epsilon_{*}}\big{)}^{n_{*}}$, where $\epsilon_{*}\equiv\epsilon(n_{*})<0$ is the per subunit energy gain upon aggregation into the optimal size. Defining the concentration scale

yields the following equation of state, relating total concentration to monomer concentration

a result derived originally by Debye to explain scattering in soap solutions Debye (1949). The dependence of the populations of monomers and $n_{*}$-mers on total concentration is plotted in Fig. 5A and can be summarized as follows. For low concentration, $\Phi\ll\phi_{*}$, and additional subunits added to the system go predominantly to monomers, since $\phi_{1}\approx\Phi$ while $\phi_{n_{*}}\approx\phi_{*}(\Phi/\phi_{*})^{n_{*}}\ll\Phi$. Notably, the population of aggregates in this regime $\phi_{n_{*}}$ is simply proportional to the random probability of $n_{*}$ free subunits to spatially coincide, $\Phi^{n_{*}}$, times the enhanced Boltzmann factor for aggregation, $e^{-n_{*}\beta\epsilon_{*}}\approx\phi_{*}^{-n_{*}}$, and hence is diminishingly small. In the large concentration regime $\Phi\gg\phi_{*}$, the dominant populations are reversed: the $n_{*}$-mer population increases in proportion to total concentration, $\phi_{n_{*}}\approx\Phi$, while monomers increase much more slowly, $\phi_{1}\approx\phi_{*}(\Phi/\phi_{*})^{1/n_{*}}\ll\Phi$. These two regimes are characterized by a crossover near $\Phi\approx\phi_{*}$, which is known as the critical aggregation concentration (CAC), although it is not strictly a phase transition for finite $n_{*}$.⁵⁵5 The CAC is commonly referred to as the critical micelle concentration (CMC) in the amphiphile literature. In the virus assembly literature it is often called the pseudo-critical concentration to emphasize that it does not correspond to a true phase transition for finite $n_{*}$, and because the CAC observed in finite-time experiments typically exceeds the equilibrium CAC due to nucleation barriers. As illustrated in Fig. 5A, the aggregation crossover becomes increasingly sharp as the aggregation number $n_{*}$ increases. Fig. Fig. 5B shows an example of CAC behavior observed in experiments of hepatitis B virus capsid protein assembly Ruan et al. (2018).

Underlying the transition is a thermodynamic trade off between translational entropy and the interaction free energy that drives aggregation. Maximizing the ideal translational entropy of aggregates favors maximizing the number of independent translational degrees of freedom, i.e. the number of independent centers of mass in the mixture. To form an aggregate, $n_{*}$ monomers must give up their $n_{*}$ centers of mass, for the single center of mass of the aggregate. Only when the aggregation free energy is sufficient to “pay” this entropic price (i.e. when the concentration of “excess” monomers is sufficiently large), does aggregation become thermodynamically favorable. Hence, the CAC depends not only on aggregation energetics, but also the aggregation number $n_{*}$. According to eq. (11), $\phi_{*}$ exhibits a modest increase with $n_{*}$ (as $\sim n_{*}^{-1/n_{*}}$) due to the increased translational entropy loss for when joining larger aggregates. We return to the implications of the $n_{*}$-dependence of aggregation thresholds in the discussion of competing aggregate states below. Notice that although the model ignores physical interactions between distinct aggregates, the change in translational entropy couples $n_{*}$ units, making aggregation a cooperative process. For this reason, the CAC becomes progressively sharper and tends toward a thermodynamic transition as $n_{*}\to\infty$ (Fig. 5A).

Here we review the conditions for the most probable, or optimal aggregate size $n_{*}$ given a known form of $\epsilon(n)$, which we assume for the moment to have a single minimum at target size $n_{\rm T}$. The optimal size $n_{*}$ corresponds to the maximum in the aggregate distribution $\rho_{n}=\big{(}e^{\beta(\mu-\epsilon(n))}\big{)}^{n}$, or equivalently, the minimum in the free energy $n\big{[}\epsilon(n)-\mu\big{]}$, which includes the total interaction free energy and entropy cost of forming an $n$-mer from free monomers. However, except under conditions where monomers are buffered to a fixed concentration, the chemical potential $\mu=\ln\phi_{1}$ varies as the equilibrium monomer population changes with total concentration. Naively, this might suggest that the optimal aggregate size should strongly vary with total concentration. Here, we illustrate why, notwithstanding the variation of $\mu$ with concentration, $n_{*}$ is nearly independent of $\Phi$ and almost entirely determined by the form of aggregation energetics, $\epsilon(n)$. Following this, we summarize the effects of dispersity (i.e. finite width of the aggregation peaks in Fig. 4B) on aggregation thermodynamics, which is necessary to account for the (weak) concentration dependence of the optimal aggregate size.

Two-state aggregation: As a first approximation, consider the two-state aggregation model deep into the aggregation regime, i.e. well above the CAC ($\Phi\gg\phi^{*}$). The optimal aggregate size derives from the condition $\frac{d\rho_{n}}{dn}\big{|}_{n_{*}}=0$, or

where $\epsilon^{\prime}=\frac{d\epsilon}{dn}$. Using the fact that $\phi_{1}(\Phi\gg\phi_{*})\simeq\phi_{*}(\Phi/\phi_{*})^{1/n_{*}}=e^{\epsilon_{*}}(\Phi/n_{*})^{1/n_{*}}$ from eq. (11) in the previous section, this transforms to the condition for the optimal (peak) aggregate size

From eq. (13) we may draw two key conclusions. First, in the limit of large target size $n_{*}\gg 1$, the optimal size corresponds to a minimum of $\epsilon(n)$. That is, since $\epsilon^{\prime}(n_{*})\to 0$, $n_{*}\to n_{\rm T}$ and the aggregate peak is selected by minimizing per sub unit aggregation energy, independent (to a first approximation) of concentration. Second, the right-hand side of eq. (13), which is proportional to the translational free energy of a dilute concentration of $n_{*}$-mers, is negative, and hence $\epsilon^{\prime}(n_{*})<0$. Combining this with eq. (12), we find the inequality,

This last condition shows that the equilibrium chemical potential approaches from below, but never quite reaches, the interaction energy of the optimal aggregate $\epsilon(n_{*})$ (excepting the unphysical limit $\Phi/n_{*}\to 1$).

Finally, the fact that $n_{*}$ corresponds to a maximum in the aggregate size distribution, suggests the following condition from $\frac{d^{2}\rho_{n}}{dn^{2}}<0$

As the righthand side goes to zero as $\sim n_{*}^{-3}$, the aggregation energetics must be convex in the vicinity of the optimal size. Strictly speaking however, a stronger condition than convexity alone is needed to justify the neglect of aggregation number fluctuations in the 2-state approximation, as discussed next.

Gaussian approximation: We now consider the effect of convexity of the aggregation energetics, characterized by the second-derivative of $\epsilon(n)$ at the peak aggregate size. As above, we restrict our analysis to the case of a single, well-defined minimum in $\epsilon(n)$ occurring at a finite target size $n_{\rm T}>1$. Close to the minimal-energy size, the energetics have the form

where $\epsilon_{\rm T}<0$ and $\epsilon^{\prime\prime}_{\rm T}>0$ respectively characterize the minimum energy and convexity of the target aggregate, as illustrated in the harmonic approximation in Fig. 4A. Physically, $\epsilon^{\prime\prime}_{\rm T}$, which we call the convexity, quantifies (twice) the energetic cost (in $k_{\text{B}}T$) to alter the aggregate number from its target by $\pm 1$. In the following section, we will describe the physical effects that control convexity in different models of self-limiting assembly. Here, we see that the concentration-dependence of the mean (or peak) self-limiting size, as well as the size-dispersity, are controlled by a single combination of $\epsilon^{\prime\prime}_{\rm T}$ and $n_{\rm T}$.

The effect of finite convexity is to allow fluctuations in $n$ around the peak size $n_{*}$. When $\epsilon^{\prime\prime}_{\rm T}$ and $n_{\rm T}$ are sufficiently large, the aggregate distribution follows a Gaussian,

where $\langle\Delta n^{2}\rangle$ characterizes the variance of aggregate sizes relative to $n_{*}$. Assuming that the Gaussian distribution of aggregates is well separated from the monomer peak, the size fluctuations around $n_{*}$ may be summed in $\phi_{n}=n\rho_{n}$, yielding the same mass-action formula as eq. (11), but with a redefined CAC,

Compared to the two-state approximation, $\phi_{*}$, is depressed by a factor proportional to $\langle\Delta n^{2}\rangle^{1/2n_{*}}$ due to the comparative increase in the number of aggregates states and associated entropy. Likewise, well above the CAC, the monomer population is depressed (relative to the 2-state approximation) by the same factor. Combining this effect into the chemical potential with the peak aggregate condition in eq. (12) gives the following prediction for peak (mean) aggregate

where we are considering only the leading correction to $n_{*}-n_{\rm T}$. In this same limit, aggregate dispersity becomes,

where again averages are taken with respect to aggregate distribution $\rho_{n}$ (i.e. number average).

The results of the Gaussian approximation in eqs. (19) and (20) highlight two physical effects of convexity. First, in eq. (19), the mean aggregate size always falls slightly below the minimal-energy target size (i.e. because $\Phi<1$, and hence, the logarithmic factor is always negative). This “sub-optimal” aggregate size derives from the (translational) entropic preference for smaller-$n$ aggregates, and hence, this weak depression of $n_{*}$ decreases with increasing supersaturation as the translation free energy of aggregates $(k_{\text{B}}T/n_{\rm T})\ln(\Phi/n_{\rm T})$ tends toward zero. Second, the relative shift of mean aggregation number $(n_{*}-n_{\rm T})/n_{\rm T}$ and the relative size variance $\langle\Delta n^{2}\rangle^{1/2}/n_{\rm T}$ decrease with the reciprocal of $\epsilon_{\rm T}^{\prime\prime}n_{\rm T}^{3}/k_{\text{B}}T$. Hence, corrections from size variations become small, in relative terms, either for sharp minima, when $\epsilon_{\rm T}^{\prime\prime}\gg k_{\text{B}}T$, or for larger aggregate number, when $n_{\rm T}\gg(k_{\text{B}}T/\epsilon_{\rm T}^{\prime\prime})^{1/3}$. While at first glance, this might suggest a generic tendencies toward monodisperse aggregation in the large $n_{\rm T}$ limit, SLA models described in Sec. III show convexity to be a decreasing function of $n_{\rm T}$. Hence, as it turns out, the decrease of relative size fluctuations with target size becomes non-trivially dependent on the geometric sensitivity of aggregation energy.

Self-limitation without minima: Before considering landscapes with more complex equilibria, we briefly note that it is possible to construct functional forms of $\epsilon(n)$ which exhibit self-limitation without local minima. As an example, consider a variation of the general type of energetics described in Sec II.1.2,

where $\Delta(n)$ is a monotonically decreasing, and convex, function of $n$ so that the minimal energy per particle occurs for infinite aggregates, i.e. $\epsilon(n\to\infty)=-\epsilon_{0}$. The condition for a maximum in $\rho_{n}$ in eq. (12) gives

Because the chemical potential is bounded from above by $-\epsilon_{0}$ (from eq. (14)) and hence $(\Delta\mu)_{\infty}>0$, the conditions for finite optimal aggregate size can be satisfied (at some accessible $\mu$) provided that $\Delta(n)$ decreases faster than $1/n$, or more specifically, that

for some range of finite $n$. For example, for any model that approaches the bulk energy as a power-law $\Delta(n)=\Delta_{0}/n^{\gamma}$ where $\gamma>1$, it is straightforward to show the peak size obeys $n_{*}=\big{[}(\gamma-1)\Delta_{0}/(\Delta\mu)_{\infty}\big{]}^{1/\beta}$, which increases continuously with concentration as $\Delta\mu_{\infty}\to 0$ from above.

While the previous argument shows that it is possible to construct mathematical examples of minima-free energy densities that result in finite-$n$ peaks in $\rho_{n}$, physical cases of SLA fall outside of this category. For example, generic physical grounds suggest that aggregates possess a boundary, or surface, at which the assembly energy is different (generally higher) than in the interior, as described for cases of short-ranged cohesive interactions in Sec.  II.1.2. In the limit of large $n$, the asymptotic contribution to the energy density from this boundary, $n^{-1/d}$ where $d\geq 1$, will dominate over other possible terms falling off faster than $n^{-1}$ (such as the $\Delta(n)$ term in eqs. (21) and (23)), leading to unlimited assembly. Hence, we exclude such anomalous cases, and focus the discussion on situations where self-limitation is directly associated with well-defined minima of $\epsilon(n)$.

Secs. II.2.1 and II.2.2 above describe the simplest case of self-limiting assembly: a concentration-controlled crossover, or “pseudo-transition”, from a monomer-dominated state to a state dominated by aggregates of one finite size. The finite aggregate size corresponds to a single minimum in $\epsilon(n)$, and the transition occurs at a single CAC. In this section, we overview the thermodynamics of cases in which assembly is characterized by multiple local minima, or instead, by transitions between self-limiting and unlimited aggregation states. In these cases, the aggregation thermodynamics can exhibit a more complex dependence on concentration, corresponding to secondary CACs between different aggregation states. While concentration-dependent transitions between different aggregates are commonly attributed to interactions between aggregates Israelachvili (2011), it is less widely appreciated that they can also occur in ideal aggregation models, which strictly neglect inter-aggregate interactions. As we review in Sec. 11, the possibility of an “ideal” secondary CAC was first considered in the context of transitions between spherical and cylindrical surfactant micelles Porte et al. (1984); May and Ben-Shaul (2001). In this section, we describe this behavior as a generic consequence of the translational entropy preferences for smaller aggregate sizes, and as such, ideal secondary CACs can occur in a much broader class of SLA models.

Two finite aggregate states: We first describe a simple model with only 2 states of finite aggregates in equilibrium with a population of free monomers, for simplicity ignoring number fluctuations around these three states. We consider two states of small and large aggregates, corresponding to two well-defined local minima of $\epsilon(n)$, at $n_{\rm S}$ and $n_{\rm L}>n_{\rm S}$ subunits, respectively, as shown schematically in Fig. 6A. In this case, aggregation is controlled by not only the difference in the respective energy minima $\epsilon_{\rm S}$ and $\epsilon_{\rm L}$, but also the difference in the aggregation number. To understand aggregation in the presence of multiple minima, it is convenient to define the nominal CACs corresponding to either aggregate state, from eq. (10),

These are concentrations at which aggregates of either type would overtake free monomers, were it not for the additional equilibrium between the populations of small and large aggregates. In terms of these concentration scales, the law of mass action takes the form

where the last two terms represent the respective populations of subunits in $n_{\rm S}$-mers and $n_{\rm L}$-mers , which we denote as $\Phi_{\rm S}$ and $\Phi_{\rm L}$. As above for the case of a single minimum in $\epsilon(n)$, in the limit of high concentration, aggregation always proceeds towards the state with the lowest energy. However, in this case, there are two possible thermodynamic scenarios for the concentration dependence, depending on the relative energy difference between small and large aggregates:

i) $\epsilon_{\rm S}<\epsilon_{\rm L}$: Because $n_{\rm S}<n_{\rm L}$, in this case it is always true that $\phi^{*}_{\rm S}<\phi^{*}_{\rm L}$, which means that as concentration increases, $\phi_{1}\to\phi^{*}_{\rm S}$ before reaching $\phi^{*}_{\rm L}$. Above the threshold where $\phi_{1}\approx\Phi_{\rm S}$ , monomers remain effectively “buffered” at $\phi_{1}\approx\phi^{*}_{\rm S}$, and it is straightforward to show that $\Phi_{\rm L}\ll\Phi_{\rm S}$ ⁶⁶6The assumptions of ideal aggregation require that $\Phi$ must remain below unity, a condition that requires $\phi_{1}\ll\phi^{*}_{\rm L}$ for case $(i)$.. Thus, when the smaller aggregate has a lower per subunit aggregation energy, aggregation proceeds as if there was only a single target state with a CAC at $\phi^{*}_{\rm S}$ and never yields a significant number of large aggregates.

ii) $\epsilon_{\rm S}>\epsilon_{\rm L}$: When the large aggregates are energetically favored, there are two possibilities. Consider first the case of large energy differences, such that $e^{\beta(\epsilon_{\rm L}-\epsilon_{\rm S})}<\frac{n_{\rm L}^{1/n_{\rm L}}}{n_{\rm S}^{1/n_{\rm S}}}$. For this first regime $\phi^{*}_{\rm L}<\phi^{*}_{\rm S}$, and there is only a single CAC at the (lower) critical concentration for large aggregates. For the second regime, when the energy difference between large and small aggregates is smaller, in the range $1>e^{\beta(\epsilon_{\rm L}-\epsilon_{\rm S})}>\frac{n_{\rm L}^{1/n_{\rm L}}}{n_{\rm S}^{1/n_{\rm S}}}$, the order of the CACs reverses: $\phi^{*}_{\rm S}<\phi^{*}_{\rm L}$, and leads to two CACs. As shown shown in Fig. 6B, for this case upon increasing concentration from the dilute limit, the concentration reaches a first CAC at $\Phi\approx\phi^{*}_{\rm S}$, with a transition to a state dominated by small aggregates (i.e. $\Phi_{\rm S}\gg\phi_{1},\Phi_{\rm L}$). This state persists until reaching a second CAC at $\Phi\approx\phi^{**}$, defined by a crossover in dominant aggregation state to $\Phi_{\rm L}>\Phi_{\rm S}$. The concentration threshold condition can be estimated by solving for the monomer concentration at which $\Phi_{\rm S}(\phi_{1})=\Phi_{\rm L}(\phi_{1})\gg\phi_{1}$ from eq. (25)

which is larger than the “bare” value $\phi_{\rm L}^{*}$ owing to the depletion of free monomers by small aggregates, and corresponds to a total concentration $\Phi^{**}\approx 2\phi^{**}$. The high-concentration regime above the second CAC is dominated by minimal-energy large aggregates, but maintains a sizeable amount of small aggregates (i.e. $\Phi_{\rm L}\gg\Phi_{\rm S}\gg\phi_{1})$, approximately buffered at the second CAC concentration.

To summarize this 2-aggregate model, it is possible to have two pseudo-critical transitions (as in Fig. 6B), first from disassembled monomers to small aggregates and then from small to large aggregates, provided the energy difference between aggregates is sufficiently small, in the window

This is consistent with the secondary CAC behavior shown in the assembly state diagram in Fig. 6C, calculated for the case of $n_{\rm S}=10$ and $n_{\rm L}=50$.

The physical origin of this double CAC behavior can be traced to a competition between the higher cohesive energy of large aggregates pitted against the higher (per subunit) translational entropy of smaller aggregates. This can be cast in terms of the chemical equilibrium between large and small aggregates, which requires

Energetically favorable large aggregates, $\epsilon_{\rm L}<\epsilon_{\rm S}$, require aggregate concentrations to adjust to maintain a suitably higher translational entropy of small aggregates, namely $\frac{1}{n_{\rm S}}\ln\rho_{\rm S}<\frac{1}{n_{\rm L}}\ln\rho_{\rm L}$, specifically

This condition shows that the larger entropy of smaller aggregates requires that $\rho_{\rm S}/\rho_{\rm L}>1$ provided that the concentration of large aggregates remains sufficiently small, below $\rho_{\rm L}<\rho_{\rm L}^{**}=\Big{[}e^{\beta(\epsilon_{\rm L}-\epsilon_{\rm S})}\Big{]}^{n_{\rm S}n_{\rm L}/(n_{\rm L}-n_{\rm S})}$. Hence, when $n_{\rm S}$ and the differential in aggregation energy are small enough, small aggregates remain more populous than large aggregates up to total concentrations that exceed the first CAC to the small aggregate state, until the second CAC.

This simplified model illustrates a generic conclusion. Even if an aggregate state does not correspond to the global minimum of $\epsilon(n)$, it may exhibit an entropically-stabilized window of thermodynamic dominance at intermediate concentrations, provided its target size is sufficiently small and its energy is sufficiently close to the global minimum. Next, we illustrate this entropic stabilization of finite (compact) aggregates in models for which the competing states are unlimited.

Finite and unlimited aggregates: While polymorphic assembly into multiple finite-number aggregates occurs in some natural and biomimetic systems Wingfield et al. (1995); Lutomski et al. (2018); Sun et al. (2007) and may be desirable for nanomaterials applications, cases in which aggregates change dimensionality are more common. That is, aggregate structures that remain finite in at least one or more spatial directions, but undergo essentially unlimited growth in other directions. The most common examples are amphiphillic assemblies, which can either form spherical micelles (finite in all directions), cylindrical micelles (finite in two spatial dimensions, unlimited in one), or lamellar/layered assemblies (finite in one dimension, unlimited in two). In Sec. 11 below we describe the molecular ingredients that lead to polymorphic transitions between aggregate dimensionality based on a model of surfactant assembly May and Ben-Shaul (2001); Bergström (2016). In this section, we illustrate how the principles of secondary CAC behavior apply to models which can exhibit states of finite aggregation number (e.g. spherical) that can transition to states of 1D aggregation or a bulk (unlimited) morphology.

Since our primary interest is to describe conditions where ideal aggregation gives rise to concentration-dependent transitions in morphology, we consider a minimal description of a generic model including finite and unlimited aggregation states. As summarized in Fig.7A, this model considers three disconnected “branches” of assembly:

1) Finite (compact) aggregates, with respective target size and energy, $n_{\rm F}$ and $\epsilon_{\rm F}$

2) 1D aggregates, with energy per subunit

where $\Delta_{0}>0$ characterizes the cost of finite “endcaps” and $\epsilon_{\rm 1D}<0$ is the limiting $n\to\infty$ per subunit assembly energy in this morphology

3) Bulk aggregates, with energy density $\epsilon_{\text{bulk}}$. Here, we consider only one macroscopic aggregate ($n\to\infty$) with negligible boundary energy.

Based on the foregoing analysis of the 2-finite assembly state model, it can be anticipated that secondary CAC behavior from finite to 1D aggregation takes places when the $n\to\infty$ energy density of 1D aggregates is lower than finite aggregates, but the energy gap is sufficiently small that the translational entropy associated with the compact aggregates can stabilize a window of $n_{\rm F}$-mer aggregates. Likewise, when the energy density of the bulk state falls below these dimensionally limited states, we anticipate an upper limit to concentration (i.e. saturation) which can maintain equilibrium with dispersed aggregates.

With this in mind, we consider a simplified law of mass action for subunit populations

where the terms respectively describe the populations of free monomers, subunits in a single finite aggregate size, 1D aggregates of various size, and bulk aggregation. The population of subunits in finite aggregates is given by (neglecting number fluctuations)

while the 1D aggregate population is given by

In this final term, we made the additional assumptions that 1D assembly is not favorable below some aggregate size close to $n_{\rm F}>1$, and that the monomer concentration remains well below the value $e^{\beta\epsilon_{\rm 1D}}$ where $\Phi_{\rm 1D}$ diverges⁷⁷7 Specifically, we assume $n_{\rm F}>e^{\beta\delta}$. For $n_{\rm F}<e^{\beta\delta}$, $\Phi_{\rm 1D}\simeq e^{-\beta\Delta}\big{(}\phi_{1}e^{-\beta\epsilon_{\rm 1D}}\big{)}^{n_{\rm F}+1}\big{(}1-\phi_{1}e^{-\beta\epsilon_{\rm 1D}}\big{)}^{-2}$, $\phi_{1}^{**}\approx e^{\beta\epsilon_{\rm 1D}}\big{(}1-e^{-\beta\delta/2}/n_{\rm F}^{1/2}\big{)}$.. The results are not qualitatively sensitive to these approximations.

Based on these forms, it is straightforward to find the free monomer concentration $\phi_{1}^{**}$ where finite aggregates and 1D aggregates are equally populous, i.e. $\Phi_{\rm F}(\phi_{1}^{**})=\Phi_{\rm 1D}(\phi_{1}^{**})$,

where

is the energy difference, or “barrier”, between an $n_{\rm F}$-mer and a 1D aggregate of the same size (see Fig. 7A). As described above, a stable aggregate population requires at least a local minimum in the energy and hence a barrier necessarily separates aggregation states associated with distinct local maxima in population. Critically, the size of this barrier determines the window of secondary CAC transition behavior, as follows.

First, note that $\delta>0$ implies that the energy of forming two “end caps” on the 1D aggregate exceeds that of the target $n_{\rm F}$-mer. Second, the existence of a second CAC requires that this concentration exceeds the primary CAC to a $n_{\rm F}$-mer dominated state, that is, the condition $\phi_{1}^{**}>\phi_{\rm F}^{*}\simeq e^{\beta\epsilon_{\rm F}}/n_{\rm F}^{1/n_{\rm F}}$. This gives an upper limit to the energy gap between $n_{\rm F}$-mer aggregation and 1D assembly for second CAC behavior, $\epsilon_{\rm F}-\epsilon_{\rm 1D}<{\Delta\epsilon_{\rm max}}$, with

, the stability window of the $n_{\rm F}$-mer state expands, in terms of $\Delta\epsilon_{\text{max}}$, both with decreasing target size and increasing energy barrier to 1D aggregation. For energy gaps larger than this limiting condition, the intermediate $n_{\rm F}$-mer state disappears ⁸⁸8 Note that eq. (35) is an implicit relation for ${\Delta\epsilon_{\rm max}}$, since $\delta$ is a function of $(\epsilon_{\rm 1D}-\epsilon_{\rm F})$. However, in the limit $\delta\gg 1$, the maximum gap is approximately ${\Delta\epsilon_{\rm max}}\approx\left(\epsilon_{\rm 1D}+k_{\text{B}}T\ln n_{\rm F}\right)/{n_{\rm F}}$. Similarly, in the limit $n_{\rm F}<e^{\beta\delta}$, ${\Delta\epsilon_{\rm max}}\approx\left(\Delta_{0}-2k_{\text{B}}T\right)/n_{\rm F}$ (see previous footnote 7)..

Last, note that monomers reach chemical equilibrium with the bulk state at a concentration $\phi_{1}=e^{\beta\epsilon_{\text{bulk}}}$, which sets an additional condition for second CAC behavior, $\phi_{1}^{**}<e^{\beta\epsilon_{\text{bulk}}}$. This gives the upper limit to the energy gap between 1D and bulk assembly for second CAC behavior before saturation,

This condition shows that concentration range of the 1D aggregate state diminishes with increasing energy barrier between compact and 1D aggregates.

An example assembly state diagram for this model under conditions $\epsilon_{\rm F}>\epsilon_{\rm 1D}>\epsilon_{\text{bulk}}$ is shown in Fig. 7B. The concentration dependent state of aggregation is plotted versus energy gap between $n_{\rm F}$-mers and 1D aggregates for variable $\epsilon_{\rm F}$ with fixed $n_{\rm F}$, $\Delta$, $\epsilon_{\rm 1D}$, and $\epsilon_{\text{bulk}}$. Under these conditions, when the aggregation energy of (compact) $n_{\rm F}$-mers is larger than, but sufficiently close to, that of (infinite) 1D aggregation, the system undergoes a sequence of concentration-driven transitions: first from monomers to finite-aggregations; then to 1D aggregates; and finally, to bulk (unlimited) assembly. In the following section, we revisit the possibility of multi-CAC behavior in the context of polymorphism of surfactant aggregates, focusing on its microscopic origin for this effect in terms of an underlying molecular model of aggregation.

We first review the case of tubule or shell-like assemblies. Examples of these are common in biology, where “tapered” protein subunits select a preferred radius of assembly curvature Oosawa and Asakura (1975). Quasi-cylindrical (tubular) examples include microtubules Nogales (2000); Cheng et al. (2012) and the bacterial flagella  Namba and Vonderviszt (1997), while quasi-spherical (shell and capsule) examples include clathrin cages Kirchhausen et al. (2014); Mettlen et al. (2018); Bucher et al. (2018); Giani et al. (2017), viral capsids Zlotnick and Mukhopadhyay (2011); Mateu (2013); Hagan (2014); Perlmutter and Hagan (2015); Bruinsma and Klug (2015); Hagan and Zandi (2016); Twarock et al. (2018); Zandi et al. (2020), bacterial microcompartments Kerfeld et al. (2010); Schmid et al. (2006); Iancu et al. (2007); Rae et al. (2013); Bobik et al. (2015); Chowdhury et al. (2014); Kerfeld and Melnicki (2016), and other protein-shell organelles Nott et al. (2015); Zaslavsky et al. (2018); Pfeifer (2012); Sutter et al. (2008). Recently, engineered tapered or patchy colloids have also drawn interest for their ability to realize synthetic analogs of self-closing tubules and shells Li et al. (2011); Morphew and Chakrabarti (2017), although many if not most such realizations to date are properly categorized as analogs to micelles where curvatures are comparable to colloidal dimension. Whatever the underlying subunit structure, the existence of 1D curvature only ensures equilibrium self-limitation along one of the two assembly dimensions in the tubular constructs, while the preferred positive Gaussian curvature of shells and capsules leads to equilibrium states with finite subunit number.

To illustrate the self-limiting thermodynamics of shells and capsules, where target curvature radii are much larger subunit dimensions, consider the following minimal model. Spherical fluid capsules, shown schematically in Fig. 9A, are composed of subunits with nominal area $a_{0}$ and a tapered shape that favors a preferred (target) spherical curvature radius $R_{\rm T}$. Here, we restrict the analysis to cases in which the curvature preference sufficiently disfavors locally anisotropic curvature Lázaro et al. (2018a) to limit incomplete assembly to cap-like aggregation states with positive Gaussian curvature. Competition between incomplete assemblies with positive and zero Gaussian curvature was recently considered in Ref. Mendoza and Reguera (2020). We assume the cap covers an axisymmetric “cap” domain of radius $R$ from its pole up to the aperture angle $\Theta$ (see Fig. 9A). While a closed capsule with a preferred curvature $R_{\rm T}$ has a target aggregation number $n_{\rm T}=4\pi R_{\rm T}^{2}/a_{0}$, a capsule may realize a different aggregation number $n=2\pi R^{2}(1-\cos\Theta)$, provided that either it is open (i.e. $\Theta\neq\pi$) or is deformed from its preferred taper (i.e. $R\neq R_{\rm T}$).

Taking the simplest possible model, we assume that the intra-shell order is fluid-like, such that the only elastic penalty derives from bending deformations away from target curvature, which we consider via a membrane-like bending energy

where $B$ is a bending modulus and the area integration is carried out over the incomplete shell. Additionally, we consider the line energy associated with open edge for an incomplete cap,

where $\lambda$ is the energy per unit length of the exposed edge, associated with the fewer cohesive bonds as well as a difference in solvation of subunits at the edge. The aggregation energy as a function of $n$ then takes the form

where $-\epsilon_{\rm T}$ is the “bare” aggregation energy for subunits in the bulk of undeformed capsules.

It is easy to see that the form of $\epsilon(n,\Theta)$ in eq. (39) has a global minimum for closed capsules of the target size (i.e. $n=n_{\rm T}$ and $\Theta=\pi$). Nevertheless, the combination of bending elasticity and the edge energy of incomplete shells influences assembly for $n\neq n_{\rm T}$. This can be seen by plotting the landscape of assembly energetics in the $n/n_{\rm T}$ - $\Theta$ plane, as shown in Fig. 9B. For very small aggregate sizes $n/n_{\rm T}\ll 1$, caps lock into the preferred curvature, $R\to R_{\rm T}$, described by the condition $\cos\Theta\to 1-2n/n_{\rm T}$, shown as a dotted line in Fig. 9B. As $n$ increases, the edge energy favors compression of the open caps to smaller curvature radii $R<R_{\rm T}$. The amount of this shape compression grows up to a critical aggregation number $n_{\rm S}$, beyond which the minimal energy capsule “snaps” discontinuously to a closed shell of suboptimal size, that is $\Theta\to\pi$ for $n_{\rm S}<n<n_{0}$.

Hence, the minimal aggregration vs. $n$ generically exhibits two branches (Fig. 9C): an open cap for $n<n_{\rm S}$ and an (edge-free) closed shell for $n_{\rm S}<n<n_{\rm T}$. The transition between these two branches can be understood in terms of “nucleation” of an open pore in an over-curved shell, where $n_{\rm T}-n_{\rm S}$ corresponds to the size of the “critical nucleus”. The inset of Fig.9C shows that the critical preclosure size $n_{\rm S}$ generically decreases with an increasing (dimensionless) ratio of edge energy and bending stiffness, $\bar{\lambda}\equiv(a_{0}n_{\rm T}/8\pi)^{1/2}\lambda/B$.

Notwithstanding the generic existence of a transition between open-cap and preclosed branches of the energy landscape, the transition does not lead to stable partial shells (i.e. a minima in $\epsilon(n)$ for $n\neq n_{\rm T}$). Hence, while the open cap branch and its transition to the closed shell may have implications for assembly pathways and kinetics, the equilibrium distribution of self-limiting capsules is independent of the edge energy and generically governed by the energetics of the closed-shell branch. This fact has further generic consequences for the concentration dependence and dispersity of aggregate size, both of which are governed by the product of the convexity and cube of the target size, $\epsilon^{\prime\prime}(n_{\rm T})n_{\rm T}^{3}/k_{\text{B}}T$, according to eqs. (19) and (20). The $\Theta\to\pi$ limit of (39) shows that $\epsilon^{\prime\prime}(n_{\rm T})=B\pi/n_{\rm T}^{3}$. Hence, the decrease of convexity with target assembly size precisely cancels that entropic factor of $n_{\rm T}^{3}$, such that the relative shift in mean aggregate size and relative dispersity, $(n_{*}-n_{\rm T})/n_{\rm T}$ and $\langle\Delta n^{2}\rangle^{1/2}/n_{\rm T}$, respectively, are limited only by the ratio of bending modulus to thermal energy, $B/k_{\text{B}}T$, and independent of self-closing target size. ⁹⁹9This continues until the asymptotic limit of zero spontaneous curvature ($n_{\rm T}\to\infty$), at which point the free energy per subunit is independent of size and the size distribution becomes an unlimited exponential Helfrich, W. (1986) as shown for 1D assemblies in section II.1.2. Therefore, to regulate the self-limiting size of self-closing assemblies in absolute terms, the rigidity of subunits and their angular interactions must grow with target size.

The predicted growth of size fluctuations with target size would seem to contradict observations of the best studied example of self-closing shells, icosahedral virus capsids. At conditions of optimal assembly, size polydispersity of viruses is remarkably small. In fact, this high degree of monodispersity has been exploited by using 3D crystalline arrays of virus capsids for optical applications requiring precise spatial periodicity Young et al. (2008); Steinmetz et al. (2011); Minten et al. (2011); Dang et al. (2011); Judd et al. (2014); Park et al. (2015); Malyutin et al. (2015); Delalande et al. (2016); Rother et al. (2016); Chen et al. (2016); Brillault et al. (2017). While high-precision measurements of size-dispersity in capsids are challenging, electron microscopy structures that were obtained without the assumption of icosahedral symmetry show that as many as 40% of alphavirus nucleocapsid core particles exhibit defects Wang et al. (2015, 2018), and hence some dispersity in shape. Notably some non-icosahedral structures, like immature HIV caspids, exhibit variations in number of subunits ($\sim 1000$ GAG protein subunits) that are on the order of the mean capsid size Briggs et al. (2009), although such effects may also be attributed to assembly kinetics Dharmavaram et al. (2019). More recently, size distributions of hepatitis B virus (HBV) capsids (and capsids of other viruses) have been achieved near or at single-subunit precision using resistive pulse sensing   Zhou et al. (2011, 2018), mass spectrometry   Uetrecht et al. (2011), and charge detection mass spectrometry  Pierson et al. (2014, 2016); Lutomski et al. (2018). Although metastable defective capsids are observed Pierson et al. (2016); Lutomski et al. (2018), these measurements show that at long times (potentially corresponding to a near equilibrium state) the population is dominated by icosahedral capsids with the native size ¹⁰¹⁰10In fact HBV is dimorphic. Both in vitro and in vivo HBV capsid assembly yields mostly 120 protein dimer capsids with $T=4$ icosahedral symmetry in the Caspar Klug nomenclature (see section V.2), but also a few percent of $T=3$ icosahedral capsids. However, size fluctuations around the dominant $T=4$ population were shown to be insignificant at long times..

To place these measurements in the context of the results for the fluid shell model, we note that bending moduli for virus capsids have been estimated from the force-displacement curves measured in nanoindentation experiments in which virus capsids are compressed using an AFM tip ¹¹¹¹11It should be noted that estimating elastic moduli from the force-displacement curve is sensitive to the value chosen for thickness of the capsid shell, and the relationship between the atomic structure and the effective mechanical thickness remains at least somewhat obscure May et al. (2011); May and Brooks (2011).. Estimated bending modulus values vary from 10-200 $k_{\text{B}}T$, but typical values fall in the higher end of the range, 100-200 $k_{\text{B}}T$. (For comparison, bending moduli of lipid bilayer membranes are typically in the range of $10-20$ $k_{\text{B}}T$.) Using the result from above that $\langle\Delta n^{2}\rangle^{1/2}\approx n_{0}/\sqrt{\pi B}$, with $n_{0}=120$ and an estimate of $B\thicksim 60k_{\text{B}}T$ for HBV¹²¹²12The bending modulus is calculated from the 3D Youngs modulus $E=0.26$ GPa obtained from nanoindentation measurements in Roos et al.Roos et al. (2010), and assuming a thin shell model so that the 2D Youngs modulus and bending modulus are respectively given by $Y=Et$ and $B=Yt^{3}/(12(1-\nu^{2}))$, with $t=2.1$ nm the effective shell thickness Roos et al. (2010); Wynne et al. (1999) and $\nu=0.4$ the Poisson’s ratio Uetrecht et al. (2008); Roos and Wuite (2009). gives a root mean squared size fluctuation of about 9 dimers, considerably larger than the long-time experimental estimates.

This discrepancy highlights two physical ingredients neglected in the model described above: the discrete subunit size and the fact that most virus capsids, as well as most other protein shells, are crystalline rather than fluid. As discussed below in section V.2, the arrangement of proteins within icosahedral capsids can be mapped onto a triangular net. However, tiling a spherical topology with a triangular lattice requires the formation of 12 five-fold sites, often consider as “defects” in a hexagonal packing. A number of equilibrium calculations have shown that the elastic energy of the defects themselves and inter-defect elastic interactions significantly affect the energy landscape of such shells Chen et al. (2007); Bruinsma et al. (2003); Zandi et al. (2004); Li et al. (2018, 2019); Mendoza and Reguera (2020). These effects are reflected in local minima in the aggregate energy at certain ‘magic numbers’ of subunits Zandi et al. (2004). Notably, these minima correspond to shells with high degrees of symmetry, with the in-plane bonding energies corresponding to shells with icosahedral symmetry. Thus, when including the energetics of this bond-ordering a size fluctuation of even one subunit can incur a significant energy cost ($\gtrsim 10k_{\text{B}}T$), since it requires disruption of the low-energy geometry, for example, through the introduction of a pair of 5- and 7-fold defects. In fact, the metastable structures observed in HBV capsids are typically found at discrete intervals corresponding to deviations of multiple subunits from the native capsid size Pierson et al. (2014, 2016); Lutomski et al. (2018), suggesting that typical fluctuations correspond to insertion or deletion of multi-subunit oligomers (e.g. hexamers of the capsid protein) which would minimize disruptions to the capsid symmetry. For example, Fig. 10A,B show cryo-EM images and examples of 2D class averages respectively of in vitro assembly products of woodchuck hepatitis B virus (WHV) capsid proteins, which exhibit heterogeneous structures including icosahedral capsids, elongated closed shells, and shells with overlapping edges. Fig. 10C shows a hypothetical model of a non-icosahedral closed shell, in which insertion of additional hexamers leads to a prolate structure.

Computational models of icosahedral assembly have also identified ensembles of defective capsules in which additional hexamers were added to the icosahedral shell Nguyen et al. (2009); Elrad and Hagan (2010), although these models and the corresponding defective structures differ from the HBV system. Hence, for more realistic models that incorporate both bending and “bond-network” elasticity, we expect that these corrugations in the energy landscape (i.e. due to communicability with icosahedral symmetry) versus $n$ will be superposed on the smooth landscapes illustrated in Fig. 9 for the fluid shell model, which may account for size fluctuations to be restricted for a limited set of low-energy values of $n$ at or near to high-symmetry, or magic number capsomer arrangements.

Arguably the most common and well-studied class of self-limiting assemblies is amphiphiles. In the broadest sense, these refer to subunits with chemically dissimilar ends, which consequently favor distinct solvent environments. For example, lipids and surfactants possess oily hydrocarbon tails attached to a polar or charged head group Israelachvili (2011), which imparts a respective hydrophobic and hydrophillic character to either end of the same molecule (e.g. schematic in Fig. 11A). Dispersing such amphiphiles in a solvent that has higher affinity to one end of the molecule generically drives them to form aggregates that partially hide, or sequester, the solvophobic portions while maintaining exposure of the solvophillic portions. Examples of such aggregrates, spherical and cylindrical micelles, and bilayer sheets, are shown schematically in Fig.  11B-C. Notably these structures curve upon themselves, but do so on a lengthscale that is limited by, and comparable to, the size of the amphiphile itself, e.g. the molecular tail length in Fig.  11A. The tendency to exclude unfavorable solvent from the “core” of the aggregate, in combination with the packing constraints of filling this region with the solvophobic portions, requires each amphiphillic subunit to “span” the entire thickness of the aggregate, which is fundamental to their self-limiting assembly.

In this section we describe a simple model to capture the self-limiting assembling of amphiphiles, and highlight, in particular, how thermodynamic considerations of changes in aggregate thickness shape the preferred aggregate curvature, but also give rise to polymorphism in the dimensionality of aggregates (e.g. spheres, cylinders, membranes). For illustration, we review a model for the thermodynamics of surfactant aggregation, of the type shown in Fig. 11A, capturing central ingredients of the well-known packing model developed by Israelachvili and coworkers Israelachvili et al. (1976); Israelachvili (2011). While this model aims to capture molecular elements of low-molecular weight surfactants and lipids, the essential thermodynamic features carry over to other amphiphillic assemblies, such as block copolymers in selective solvents Halperin et al. (1992); Leibler et al. (1983); Zhang and Eisenberg (1996); Jain and Bates (2003). The thermodynamics of amphiphile aggregation incorporates three ingredients: (i) the thermodynamics of area per solvophillic head group; (ii) the thermodynamics of molecular extension; and (iii) the constraint of uniform density in the solvophobic core, which links the first two elements.

A simple model to describe the head-group energetics considers the per subunit energy to form aggregates with an area $a$ at the solvent/core interface Israelachvili et al. (1976); Israelachvili (2011). The generic tendency to “hide” amphiphiles from the solvent is parameterized by a surface energy cost $\gamma a$, where $\gamma>0$ favors dense lateral packing of head groups. Competing against this lateral compression is the cost of inter-unit repulsive interactions, which in the simplest case are described by the two-body term in the virial series, giving a per subunit energy $A_{2}/a$, where $A_{2}>0$ Tanford (1974). These two terms can be combined into a single form

where $a_{0}=\sqrt{A_{2}/\gamma}$ is the optimal head group area.

In combination with the tendency to achieve optimal head group area are additional thermodynamics of tail packing in the core, and the costs to extend its length, $\ell$. There are various models proposed for this effect, including a finite-maximum extension Israelachvili et al. (1976); Israelachvili (2011) or instead treating the core as melt of flexible polymers Dill and Flory (1980); Ben-Shaul et al. (1984); Nagarajan and Ruckenstein (1991); Nagarajan (2002). Here, we adopt a simplified model used by May and Ben-Shaul for the free energy of tail length $\ell$ that spans from the solvent/core interface into the center of the aggregate

where $k$ is an elastic constant for intra-subunit stretch and $\ell_{0}$ is a preferred length, which parameterizes the free energy cost of deformations from a preferred conformation state of the short tail. Here, we consider $k$ and $\ell_{0}$ as a minimal description of the extensional thermodynamics, and like $\gamma$ and $a_{0}$, these parameters can be varied through a combination of subunit structure and physical-chemical conditions, e.g. temperature and solvent properties.

Extensional energetics are linked by packing constraints associated with occupying the core with a fixed density of solvophobic portions of the subunits Israelachvili et al. (1976); Israelachvili (2011). These constraints vary with the dimensionality of the limited directions in the aggregate: $d_{\rm L}=3$, spherical micelles; $d_{\rm L}=2$, cylindrical micelles; and $d_{\rm L}=1$ planar bilayers ¹³¹³13Note that $d_{\rm L}$ refers to the number of limited directions, while we use $d$ to refer to the dimensionality of the unlimited directions, e.g. $d=1$ and $d=2$ for cylindrical and lamellar aggregates.. By considering the ratio between the core volume and interfacial area of an aggregate of thickness (radius) $\ell$, it is straightforward to show that the (solvophobic) volume per subunit satisfies

As aggregates change their shape and number, uniform density requires adjustment of $a$ and $\ell$ to maintain constant $v_{0}$. Using this constraint, we rewrite the assembly energy in terms of a single dimensionless thickness,

giving