The problem with making things stick
Suppose you want to make a tiny structure from many copies of the same building block. You give each block two complementary faces, so that they can join into chains. That specifies how neighbors connect. It does not yet specify how long the chains should be.
There are straightforward ways to add that information. Build along a template of the desired length. Supply different parts for different positions, including an end piece. Introduce caps to stop growth. Or make the pieces join at an angle so that the structure eventually closes into a ring. Each strategy provides a reason that assembly should end somewhere.[1]
A subtler possibility is to leave the structure open, with places where more pieces could attach, but make larger assemblies progressively less comfortable. The pieces attract one another, yet fitting together requires a compromise that becomes more expensive as their number increases. Under suitable conditions, the competition can favor a population of finite assemblies.[2]
“Stop” needs a qualification. The result need not be a collection of identical objects, each permanently frozen at a prescribed count. The goal is a preferred size distribution. How can the parts’ local interactions produce that preference—and what would allow them to evade it?
A stack that cannot quite fit
Consider a theoretical construction studied by Nabila Tanjeem and colleagues: flexible, curved plates called curvamers. Picture a thin rectangular sheet bent into a shallow segment of a cylinder. Every plate has the same thickness and the same preferred curvature. Bending it more tightly or flattening it costs energy. Its broad faces attract those of other plates.[3]
The difficulty is thickness. In a tightly nested stack, the layers cannot all follow the same-radius arc. The outside of a curve has a larger radius than the inside. For neighboring faces to fit closely over their area, successive plates must therefore take slightly different curvatures. Yet their preferred curvatures are identical.[3]
Start with a few layers. They can compromise: some bend more tightly, others flatten a little. While the stack remains near its preferred curvature, adding layers widens the range of distortions needed for the same close fit. The distortion is distributed through the stack rather than confined to the newest contact. Larger stacks can flatten more extensively, so this small-stack picture is not an all-size growth law.[3]
The term geometric frustration describes this kind of incompatibility. The plates could individually adopt their preferred shapes, or they could conform closely to their neighbors, but they cannot all do both. Here the frustration is cumulative: over the relevant range of stack sizes, growing the assembly makes the collective compromise increasingly costly. A fixed penalty for each new contact would not have the same effect.[3,4]
Meanwhile, adhesion rewards assembly. Dividing the material into many tiny stacks leaves more faces unpaired. Combining those stacks creates contacts, but requires greater distortion. This supplies two opposing reasons to change the size, rather than a single reason to keep adding pieces.
Tanjeem and colleagues analyzed that competition using elasticity theory and particle simulations. Their models have regimes with a finite energetically preferred stack size. These are theoretical and computational results, not an experiment showing that a pile of ordinary curved objects will spontaneously sort itself into equal stacks.[3]
The restriction to close-fitting layers is also doing real work. We will return to what happens when the plates are allowed to make a less exact contact. First, there is a different trap in the phrase “energetically preferred.”
A preferred size is not a forbidden next piece
Imagine a supply of 110 building blocks. One possible arrangement is eleven chains of ten; another is ten chains of eleven. The second arrangement has one additional bond overall, but it may carry more accumulated strain. The question is which arrangement has lower energy for the same amount of material.
If an N-piece chain has total energy E(N), dividing a fixed supply M into such chains gives M/N chains. Their combined energy is therefore M times E(N)/N. Ignoring temperature, interactions between chains, and leftover pieces, it is energy per piece that selects the best division—not the total energy of one chain whose size we are free to change.[2]
This distinction can produce a seemingly contradictory result. In a simple mismatch model, parameters can be chosen so that ten-piece chains have the lowest energy per piece, even though attaching an eleventh free piece to one of them still lowers the energy. Larger chains are not forbidden. What favors ten is the comparison among alternative ways of allocating the whole supply.[5]
That calculation has deliberately left out entropy. At finite temperature, a solution has many possible arrangements of its components. Joining two freely moving objects into one restricts their independent placement, producing an entropic cost of association. At sufficiently low concentration, that cost can leave most material dispersed despite attractive bonds. As concentration rises, assemblies become more abundant, and their size distribution reflects both their internal free energies and the entropy of distributing material among them.[2]
There are consequently two questions to answer. Does an assembly’s internal mechanics favor a particular scale? And, at this temperature and concentration, what mixture of monomers and differently sized assemblies results?
Neither question asks whether the next piece physically refuses to attach. Nor does an equilibrium size calculation tell us how quickly the system reaches that distribution. An observed short chain might belong to a stable population, or it might simply be growing slowly. The distinction becomes important when we turn from a model to a molecular construction.
Building the compromise from DNA
A different realization of cumulative mismatch was built by Jonathan F. Berengut and colleagues in 2020. Their units, called PolyBricks, were made using DNA origami: a long DNA strand is folded into a designed structure by many shorter strands that bind to specified portions of it. Here DNA serves as a construction material, not as a program executed by a cell.[1]
A PolyBrick is not a curved plate. It has three roughly parallel bundles of DNA. The central bundle supplies the binding faces. The two flanking bundles act as blockers and are joined to the center by flexible links, allowing them to move relative to it. The paper’s design diagram, Figure 3, makes the arrangement particularly clear.[1]
The blocking bundles extend effectively farther along the direction of growth than the central bundle. To bring the central binding faces of neighboring bricks together, the blockers must shift out of the way. In a chain, accommodating successive neighbors displaces the blocking domains in a pattern that grows toward the ends. More bricks mean a more demanding collective arrangement of the movable parts.[1]
The molecular details make this richer than a stack of rigid sliding bars. The blockers’ ends carry flexible strands made of thymine, one of DNA’s four bases. These extensions both suppress unwanted binding and create soft steric repulsion between neighboring blockers. They are not hard walls. The flexible links and thermal motion also permit configurations that a rigid mechanical drawing would miss.[1]
In the lower-strain regime examined by the authors’ molecular simulations, a major cost of assembly comes from restricting the blockers’ movement. An isolated domain can fluctuate among many positions; neighbors constrain the motions it can make. Losing accessible configurations carries an entropic free-energy cost. The relevant resistance is therefore not simply a material spring becoming increasingly stretched. It includes movable parts having progressively less freedom.[1]
This is a different role for entropy from the loss of independent motion when whole chains join. Both enter the free-energy balance, but one concerns the internal configurations of an assembly and the other concerns the distribution of assemblies through the solution.
The design supplied two adjustable features. The researchers varied the number of short complementary binding sites on the central domain to change attraction. Separately, they varied the lengths of the thymine extensions on the blockers to change their effective size and the cost of accommodating neighbors. They also constructed control bricks without the flexible blocking mechanism.[1]
Then they made the structures and measured chain lengths with transmission electron microscopy. The histograms in Figure 6 of their paper show that changing those design features changes the length distributions. Under suitable binding conditions, PolyBricks produce a peak at short finite lengths rather than the much broader distributions associated with the controls. Fitting a strain-based thermodynamic model to the measurements gave distributions with lower variance than the control fits at comparable mean lengths.[1,6]
That is the experimental accomplishment: a mechanical design encoded in the repeating unit gives control over the distribution of sizes. It is not the production of a perfectly uniform batch. The paper’s discussion of approaching identical lengths concerns possibilities in idealized models with other forms of strain accumulation, not a uniformity achieved by these experiments.[1]
The results also distinguish the molecular system from the curved-plate picture. In the principal model fitted to the PolyBrick data, the additional strain cost of extending a chain rises linearly with length, so the total strain cost grows quadratically. The short-stack elastic approximation for curvamers instead gives a cubic total bending cost. Both express an accumulating penalty, but the different geometries and molecular freedoms produce different laws.[1,3]
Finally, the experiment should not be made to prove more about equilibrium than it does. The measured distributions agreed with the authors’ equilibrium model, after a preparation involving slow cooling and a further period left to equilibrate. The authors also used molecular simulations to investigate how the blockers obstruct attachment, and discussed conditions under which a kinetic barrier could limit growth even when further association remained thermodynamically favorable.[7]
A design can influence both which lengths are favorable and how readily longer chains form. Distinguishing those effects is part of understanding how it works, not a reason to dismiss the observed control of length.
The assembly can find another way
Return to the curved plates. We asked them to maintain close contact while their shapes changed. But why must the assembly pay for bending rather than accept a less perfect contact?
A study by Kyle Sullivan, Ryan Hayward, and Gregory Grason makes this alternative explicit. Instead of forcing the neighboring faces to conform everywhere, their model allows gaps to open within the attractive contact. That loses some adhesion, but lets the plates retain shapes closer to their individual preferences.[8]
The choice is mechanical: deform the particles, or deform the contacts. If bending is expensive relative to opening those gaps, a larger stack can avoid the escalating bending penalty assumed in the close-contact picture. The gap cost can instead become a repeating, approximately size-independent expense per layer. In parts of the model’s parameter range, the finite-size preference then gives way to unlimited stacking.[8]
This does not mean any gap destroys size selection. Its cost depends on how the attraction varies with separation, as well as on particle stiffness and shape. It means that size selection must survive comparison with the arrangements the parts can actually adopt—not just those permitted in the simplest drawing.
There is another escape that leaves the primary assemblies largely intact. Suppose the parts form strongly bound chains of a preferred length. If those chains can still connect through weak contacts that interrupt the transmission of strain, the result may be a much larger object made from repeating finite subassemblies.
Michael Wang and Gregory Grason examined this possibility in a theoretical model of mismatched polybricks, inspired by the DNA construction but not identical to it. Their model allows weaker, partially bound junctions between the strongly bound sections. These junctions relieve the need to continue the same accumulated mismatch through the entire connected chain.[9]
Temperature then has two jobs. It must be low enough for the strong primary contacts to hold, yet thermal and mixing effects must still compete with the weaker secondary contacts. In the model, a window of temperature and concentration favors separate finite assemblies. Stronger aggregation outside that window can join them into longer chains. Cooling further is not automatically an improvement.[9]
This is why “the objects have a preferred size” needs an account of what counts as an object. The strongly bound sections may retain their characteristic length even while the whole connected aggregate becomes large. For an application requiring isolated particles, preserving a small internal repeat unit would not be enough.
What has been programmed?
The design problem has become more specific. It is not simply to make parts attract, or even to make them fit badly. It is to arrange a competition in which joining saves enough free energy to build something, while the cost of accommodating further neighbors favors a finite scale. That preference must then survive the available alternatives: deforming contacts, rearranging parts, and joining already formed assemblies.
For curved plates, the essential choices concern curvature, thickness, flexibility, and adhesion. For the DNA PolyBrick, they concern movable blocking domains, binding sites, and soft molecular extensions. Temperature, concentration, and the rates of rearrangement help determine what those designs actually produce.[1–3,8,9]
There is no count stored in a component that announces when the target has been reached. Instead, the physical condition of a part depends on its neighbors, and their conditions depend on further neighbors. A locally repeated interaction can therefore produce a cost that depends on the size of the whole assembly.
The achievement is not matter becoming unable to grow. It is matter designed so that making more material need not mean making proportionately larger objects. Within the right regime, a population of finite assemblies can be the preferred use of the parts.
Notes
[1] J. F. Berengut, C. K. Wong, J. C. Berengut, J. P. K. Doye, T. E. Ouldridge, and L. K. Lee. “Self-Limiting Polymerization of DNA Origami Subunits with Strain Accumulation.” ACS Nano 14(12), 17428–17441 (2020). DOI: 10.1021/acsnano.0c07696. Figure 3 shows the PolyBrick construction and control; Figure 5 concerns simulations of fixed-length chains; Figure 6 presents experimental length histograms and fitted distributions. These are distinct evidence layers. The account here is based on the main article; its separate Supporting Information has not been inspected for this article.
[2] M. F. Hagan and G. M. Grason. “Equilibrium mechanisms of self-limiting assembly.” Reviews of Modern Physics 93, 025008 (2021). DOI: 10.1103/RevModPhys.93.025008. Section II develops the equilibrium population treatment, including concentration and mixing entropy. Minimizing an internal energy per building block is an explanatory step for the mechanism discussed here, not a universal necessary criterion for every form of self-limiting assembly.
[3] N. Tanjeem, D. M. Hall, M. B. Minnis, R. C. Hayward, and G. M. Grason. “Focusing frustration for self-limiting assembly of flexible, curved particles.” Physical Review Research 4, 033035 (2022). DOI: 10.1103/PhysRevResearch.4.033035. The cubic total bending cost applies to the short-stack approximation, not arbitrarily tall stacks. The analysis also allows relaxation of overall curvature; the perfectly conformal model must not be conflated with models that permit gaps or lateral sliding.
[4] S. Meiri and E. Efrati. “Cumulative geometric frustration in physical assemblies.” Physical Review E 104, 054601 (2021). DOI: 10.1103/PhysRevE.104.054601. The distinction between cumulative incompatibility and a fixed local penalty matters here; no general classification theorem from this paper is required by the article.
[5] This is an illustrative calculation, not a fit to either physical system. Take a chain whose minimized energy is E(N) = −b(N−1) + a(N³−N), with b = 20 and a = 0.01 in a common arbitrary energy unit. This follows when harmonic units with scalar deformations must differ by a fixed mismatch between neighbors. E(N)/N is minimized at N = 10, but E(11) = −186.8 is lower than E(10) + E(1) = −170.1. For the same supply of 110 units, eleven ten-unit chains have energy −1871.1, compared with −1868 for ten eleven-unit chains. The example excludes entropy, alternative contacts, and other shapes; its size is not an experimental prediction.
[6] Measurement qualifications in Berengut et al.: approximately 73% of PolyBrick particles were included because their intended binding arrangement could be identified. Other particles were ambiguous or apparently malformed; the authors could not distinguish all possible contributions from orientation, sample preparation, and structure. The mean–variance comparison in Figure 6F is calculated from fitted distributions, not a separate measurement. The paper also discusses missed monomers, possible undersampling of long control chains, and imperfect control units. Thus the histograms are not an unqualified census of every object in solution, and the fitted free-energy parameters are inferred rather than independently measured.
[7] Berengut et al., main-text Methods and discussion preceding the Conclusions. The reported preparation includes 45 hours of cooling followed by 24 hours at room temperature. The discussion of association barriers draws on steered molecular simulations, not direct experimental measurements of association and dissociation rates. Agreement with a strain-based equilibrium model is the supported claim; this article does not infer either that kinetic effects were excluded or that the measured distributions were merely kinetic arrest.
[8] K. T. Sullivan, R. C. Hayward, and G. M. Grason. “Self-limiting stacks of curvature-frustrated colloidal plates: Roles of intraparticle versus interparticle deformations.” Physical Review E 110, 024602 (2024). DOI: 10.1103/PhysRevE.110.024602. The inspected author manuscript develops the competition between bending and cohesive-gap deformation. Its minimized stacks do not, by themselves, establish assembly kinetics from a dispersed bath.
[9] M. Wang and G. Grason. “Thermal stability and secondary aggregation of self-limiting, geometrically frustrated assemblies: Chain assembly of incommensurate polybricks.” Physical Review E 109, 014608 (2024). DOI: 10.1103/PhysRevE.109.014608. This is a theoretical model of primary assembly and weak secondary aggregation. The secondary-aggregation behavior described here is not being reported as an experimental observation in the 2020 DNA study.
Production record: Continuity proposed this topic and developed the essay from published research and an explicitly simplified teaching model. RE selected it for development, supplied the PolyBrick paper, and completed the editorial review. A separate AI instance conducted an independent scientific audit of the text and notes, including source verification and checks of the teaching calculation. The article is an explanatory synthesis, not a claim of a new physical mechanism.
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