Strict Self-assembly Research Articles

Strict Self-Assembly of Fractals Using Multiple Hands

In this paper we consider the problem of the strict self-assembly of infinite fractals within tile self-assembly. In particular, we provide tile assembly algorithms for the assembly of a Sierpinski triangle and the discrete Sierpinski carpet within a class of models we term the h-handed assembly model (h-HAM), which generalizes the 2-HAM to allow up to h assemblies to combine in a single assembly step. Despite substantial consideration, no purely growth self-assembly model has yet been shown to strictly assemble an infinite fractal without significant modification to the fractal shape. In this paper we not only achieve this, but in the case of the Sierpinski carpet are able to achieve it within the 2-HAM, one of the most well studied tile assembly models in the literature. Our specific results are as follows: We provide a 6-HAM construction for a Sierpinski triangle that works at scale factor 1, 30 tile types, and assembles the fractal in a near perfect way in which all intermediate assemblies are finite-sized iterations of the recursive fractal. We further assemble a Sierpinski triangle within the 3-HAM at scale factor 3 and 990 tile types. For the Sierpinski carpet, we present a 2-HAM result that works at scale factor 3 and uses 1216 tile types. We further include analysis showing that the aTAM is incapable of strictly assembling the Sierpinski triangle considered in this paper, and that multiple hands are needed for the near-perfect assembly of a Sierpinski triangle and the Sierpinski carpet.

Self-Assembling Rulers for Approximating Generalized Sierpinski Carpets

Discrete self-similar fractals have been used as test cases for self-assembly, both in the laboratory and in mathematical models, ever since Winfree exhibited a tile assembly system in which the Sierpinski triangle self-assembles. For strict self-assembly, where tiles are not allowed to be placed outside the target structure, it is an open question whether any self-similar fractal can self-assemble. This has motivated the development of techniques to approximate fractals with strict self-assembly. Ideally, such an approximation would produce a structure with the same fractal dimension as the intended fractal, but with specially labeled tiles at positions corresponding to points in the fractal. We show that the Sierpinski carpet, along with an infinite class of related fractals, can approximately self-assemble in this manner. Our construction takes a set of parameters specifying a target fractal and creates a tile assembly system in which the fractal approximately self-assembles. This construction introduces rulers and readers to control the self-assembly of a fractal structure without distorting it. To verify the fractal dimension of the resulting assemblies, we prove a result on the dimension of sets embedded into discrete fractals. We also give a conjecture on the limitations of approximating self-similar fractals.

Approximate Self-Assembly of the Sierpinski Triangle

The Tile Assembly Model is a Turing universal model that Winfree introduced in order to study the nanoscale self-assembly of complex DNA crystals. Winfree exhibited a self-assembly that tiles the first quadrant of the Cartesian plane with specially labeled tiles appearing at exactly the positions of points in the Sierpinski triangle. More recently, Lathrop, Lutz, and Summers proved that the Sierpinski triangle cannot self-assemble in the “strict” sense in which tiles are not allowed to appear at positions outside the target structure. Here we investigate the strict self-assembly of sets that approximate the Sierpinski triangle. We show that every set that does strictly self-assemble disagrees with the Sierpinski triangle on a set with fractal dimension at least that of the Sierpinski triangle (≈1.585), and that no subset of the Sierpinski triangle with fractal dimension greater than 1 strictly self-assembles. We show that our bounds are tight, even when restricted to supersets of the Sierpinski triangle, by presenting a strict self-assembly that adds communication fibers to the fractal structure without disturbing it. To verify this strict self-assembly we develop a generalization of the local determinism method of Soloveichik and Winfree.

Strict self-assembly of discrete Sierpinski triangles

Winfree (1998) showed that discrete Sierpinski triangles can self-assemble in the Tile Assembly Model. A striking molecular realization of this self-assembly, using DNA tiles a few nanometers long and verifying the results by atomic-force microscopy, was achieved by Rothemund, Papadakis, and Winfree (2004). Precisely speaking, the above self-assemblies tile completely filled-in, two-dimensional regions of the plane, with labeled subsets of these tiles representing discrete Sierpinski triangles. This paper addresses the more challenging problem of the strict self-assembly of discrete Sierpinski triangles, i.e., the task of tiling a discrete Sierpinski triangle and nothing else. We first prove that the standard discrete Sierpinski triangle cannot strictly self-assemble in the Tile Assembly Model. We then define the fibered Sierpinski triangle, a discrete Sierpinski triangle with the same fractal dimension as the standard one but with thin fibers that can carry data, and show that the fibered Sierpinski triangle strictly self-assembles in the Tile Assembly Model. In contrast with the simple XOR algorithm of the earlier, non-strict self-assemblies, our strict self-assembly algorithm makes extensive, recursive use of optimal counters, coupled with measured delay and corner-turning operations. We verify our strict self-assembly using the local determinism method of Soloveichik and Winfree (2007).

Can multivalency be expressed kinetically? The answer is yes.

Inspired by the concept of multivalency and in pursuit of ever more intricate artificial molecular machines, we investigated the strict self-assembly of a triply threaded two-component superbundle, starting from a tritopic receptor in which three benzo[24]crown-8 macrorings are fused onto a triphenylene core and a trifurcated trication wherein three bipyridinium units are linked 1,3,5 to a central benzenoid core. The result of the investigation was quite unexpected and surprising. It transpired that the rapid formation of a doubly threaded two-component complex was followed by an extremely slow conversion (a week at 253 K in CD3COCD3 to reach equilibrium) of this kinetically controlled product into a thermodynamically controlled one, namely a triply threaded two-component superbundle. This intriguing observation begs the question: are there instances in nature where multivalency is expressed as a kinetically controlled process, prior to an equilibrium state being reached, and if so, what are the biological implications, if any?

Dynamic covalent chemistry.

Dynamic covalent chemistry relates to chemical reactions carried out reversibly under conditions of equilibrium control. The reversible nature of the reactions introduces the prospects of "error checking" and "proof-reading" into synthetic processes where dynamic covalent chemistry operates. Since the formation of products occurs under thermodynamic control, product distributions depend only on the relative stabilities of the final products. In kinetically controlled reactions, however, it is the free energy differences between the transition states leading to the products that determines their relative proportions. Supramolecular chemistry has had a huge impact on synthesis at two levels: one is noncovalent synthesis, or strict self-assembly, and the other is supramolecular assistance to molecular synthesis, also referred to as self-assembly followed by covalent modification. Noncovalent synthesis has given us access to finite supermolecules and infinite supramolecular arrays. Supramolecular assistance to covalent synthesis has been exploited in the construction of more-complex systems, such as interlocked molecular compounds (for example, catenanes and rotaxanes) as well as container molecules (molecular capsules). The appealing prospect of also synthesizing these types of compounds with complex molecular architectures using reversible covalent bond forming chemistry has led to the development of dynamic covalent chemistry. Historically, dynamic covalent chemistry has played a central role in the development of conformational analysis by opening up the possibility to be able to equilibrate configurational isomers, sometimes with base (for example, esters) and sometimes with acid (for example, acetals). These stereochemical "balancing acts" revealed another major advantage that dynamic covalent chemistry offers the chemist, which is not so easily accessible in the kinetically controlled regime: the ability to re-adjust the product distribution of a reaction, even once the initial products have been formed, by changing the reaction's environment (for example, concentration, temperature, presence or absence of a template). This highly transparent, yet tremendously subtle, characteristic of dynamic covalent chemistry has led to key discoveries in polymer chemistry. In this review, some recent examples where dynamic covalent chemistry has been demonstrated are shown to emphasise the basic concepts of this area of science.

Synthesis, Structure, and Magnetism of a Novel Alkoxide Bridged Nonacopper(II) (Cu9O12) [3×3] Square Grid Generated by a Strict Self-Assembly Process

Synthesis, Structure, and Magnetism of a Novel Alkoxide Bridged Nonacopper(II) (Cu9O12) [3×3] Square Grid Generated by a Strict Self-Assembly Process

Synthesis, Structure, and Magnetism of a Novel Alkoxide Bridged Nonacopper(II) (Cu(9)O(12))

The nonanuclear copper complex [Cu9(2paop-H)6](NO3)12�9H2O (2paop = I) was prepd. and its crystal structure detd. The complex forms a [3 � 3] square grid of copper atoms with the ligand bonding in a heptadentate manner by five nitrogen atoms and two bridging oxygen atoms. Variable temp. magnetic susceptibility and magnetization data show that the complex displays predominantly ferromagnetic behavior at low temp., but a small intramol. antiferromagnetic exchange component may be present. [on SciFinder(R)]

A novel magnetically coupled nonamanganese(ii) 3 × 3 portcullis-like grid involving just oxygen bridges, generated by strict self assembly of the metal cation and a single heptadentate ligand

The heptadentate ligand 2POAP combines two bridging alkoxide oxygen donors with diazine and pyridine terminal nitrogen donors in a linear array with the binding sites positioned critically to accommodate three metal centres, but on reaction with Mn(ClO4)2·6H2O a strict self assembly process occurs, producing a unique homoleptic nonamanganese ‘square’ grid complex, involving six ligands.

Self-assembly of helical supramolecular lanthanide complexes

The spontaneous strict self-assembly of tailored oligo-multidentate ligands with d-transition metal ions leads to the selective formation of polynuclear double- and triple-helical supramolecular complexes. A significant improvement of the selectivity of the processes can be achieved by using segmental ligands which possess different binding units connected by spacers favouring helical coordination and pure heteropolynuclear helical complexes can be isolated under thermodynamic control. The application of these principles to 4f block ions requires the development of tridentate heterocyclic ligands designed for the synthesis of triple-helical self-assembled complexes containing lanthanides. The ligand 2,6 bis(1′-methyl-1H-benzimidazol-2′-yl)pyridine (L 4) reacts with Ln 3+ in acetonitrile to give the three successive complexes [Ln(L 4)] 3+, [Ln(L 4) 2] 3+ and [Ln(L 4) 3] 3+. The three ligands in [Ln(L 4) 3] 3+ are wrapped around Eu(III) and produce a pseudo- D 3 symmetrical structure considered as a building block for triple-helical luminescent supramolecular structures. The introduction of two tridentate binding units into the oligomultidentate ligand L 6 leads to the formation of dinuclear self-assembled triple-helical complexes [Ln 2(L 6) 3] 6+ which work as molecular light-converting devices on the nanometric scale. The photophysical properties together with the formation of analogous heterodinuclear and homotrinuclear complexes are discussed.

A molecular approach to solid-state synthesis: prediction and synthesis of self-assembled infinite rods

Strict self-assembly of complementary dianionic and dicationic tetrahydrogen bond donors and acceptors leading to infinite molecular chains in the solid state is achieved in aqueous solution using both directionally controlled hydrogen bonding and ion-pairing electrostatic interactions.

Self-assembly and photophysical properties of lanthanide dinuclear triple-helical complexes

The dinucleating ligand bis[ 1- methyl-2-(6'- [ 1”-(3,5 imethoxybenzyl)benzimidazol-2~'-ylp]y rid-2'-y1)benzimidazol- 5-yllmethane (L) reacts with lanthanide perchlorates to give dinuclear 2:3 complexes [Lnz(L)#+ (Ln = La, Eu, Gd, Tb, and Lu). Detailed ES-MS, lH-NMR, luminescence, and spectrophotometric measurements in acetonitrile show that the cations [Ln2(L)#+ are produced by strict self-assembly and adopt a triple-helical structure in solution (pseudo-D3s ymmetry). The crystal structure of [EU~(L)~] C~O~)~((1C1H, E~UC2NC1)9s~H 177N39036_Cl6,a = 17.634- (3) A, b = 21.408(4) A, c = 29.437(7) A, a = 82.13(1)', /3 = 85.76(1)', y = 89.79(1)', triclinic, P1, Z = 2) shows a dinuclear pseudo-D3 triple- elical cation, [Eu2(L)#+, where the three bis(terdentate) ligands L are wrapped around the helical axis defined by the europium atoms. The Eu(II1) of each site is 9-coordinated by six nitrogen atoms of the benzimidazole units occupying the vertices and three nitrogen atoms of the pyridine units occupying the capping positions of a slightly distorted, tricapped trigonal prism. Luminescence studies of the crystalline complex [Eu~(L)~]- (ClO.n n = 2, solv = H20,6; n = 9, solv = H20,7) confirm the pseudo-D3 symmetry of the Eu(II1) sites in 11 and show that secondary interactions with water molecules in 6 and 7 destroy the trigonal symmetry. An efficient intramolecular energy transfer between the %A* excited state centered on L and the excited levels of Eu(II1) and Tb(II1) is observed (antenna effect) together with a dipole4ipolar Tb - Eu intramolecular energy transfer in the heterodinuclear-doped Eu-Tb compound. Stability constants and 1H NMR in acetonitrile show that the homodinuclear complexes [LI I ~ (L) ~ ]a~re+ l ess stable for the heavier lanthanides Tb and Lu. The origin of this effect is discussed together with the nonstatistical distribution of the different species observed when stoichiometric quantities of L (3 equiv) are mixed with Ln1(C104)3 (1 equiv) and L I I ~ ( C ~ O(1~ e)q~ui v) in solution (Ln1 # Ln2; Ln1 = La, Eu, Tb, and Lu; Ln2 = Tb and Lu).

Supramolecular chemistry of [M(CO)3(µ3-OH)]4(M = Mn, Re): spontaneous strict self-assembly of distorted super-diamondoid networks that are capable of enclathrating acetonitrile

Cocrystallization of [M(CO)3(µ3-OH)]4(M = Mn, Re) with the linear difunctional H-bond acceptor molecules 1,4-diaminobenzene and 4,4′-bipyridyl affords, respectively, twofold and fourfold tetragonally distorted super-diamondoid networks that contain microchannels which enclathrate acetonitrile.