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Rational design of self-assembly pathways for complex multicomponent structures

William M. Jacobs, Aleks Reinhardt, Daan Frenkel

arXiv:1502.01351v1cond-mat.softcond-mat.mtrl-sci

TL;DR

The paper develops a predictive theory to explain the success of DNA-brick self-assembly. It finds that temperature, coordination number, and bond-energy distributions shape nucleation, and that experimental conditions may need to change during assembly.

Abstract

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The field of complex self-assembly is moving toward the design of multi-particle structures consisting of thousands of distinct building blocks. To exploit the potential benefits of structures with such `addressable complexity,' we need to understand the factors that optimize the yield and the kinetics of self-assembly. Here we use a simple theoretical method to explain the key features responsible for the unexpected success of DNA-brick experiments, which are currently the only demonstration of reliable self-assembly with such a large number of components. Simulations confirm that our theory accurately predicts the narrow temperature window in which error-free assembly can occur. Even more strikingly, our theory predicts that correct assembly of the complete structure may require a time-dependent experimental protocol. Furthermore, we predict that low coordination numbers result in non-classical nucleation behavior, which we find to be essential for achieving optimal nucleation kinetics under mild growth conditions. We also show that, rather surprisingly, the use of heterogeneous bond energies improves the nucleation kinetics and in fact appears to be necessary for assembling certain intricate three-dimensional structures. This observation makes it possible to sculpt nucleation pathways by tuning the distribution of interaction strengths. These insights not only suggest how to improve the design of structures based on DNA bricks, but also point the way toward the creation of a much wider class of chemical or colloidal structures with addressable complexity.

STRUCTURE CONNECTIVITY DETERMINES ASSEMBLY

The paper represents a target structure as a connectivity graph and uses its connected fragments to calculate how assembly free energy and nucleation depend on structure and temperature.

STRUCTURE CONNECTIVITY DETERMINES ASSEMBLY

ASSEMBLY REQUIRES A TIME-DEPENDENT PROTOCOL

Stable partial structures can be favored over the complete target while nucleation is rate-controlling. The analysis and simulations indicate that assembly therefore requires nucleation at higher temperature followed by further cooling.

ASSEMBLY REQUIRES A TIME-DEPENDENT PROTOCOL

  • At 319 K, the 86-particle target has a stable partial intermediate of about 65 particles, favored by entropy over the unique complete structure.The critical nucleus has V‡ = 8, one fewer than the nine subunits needed to form a bicyclic fragment.Source: ASSEMBLY REQUIRES A TIME-DEPENDENT… · p. 3
  • Stable intermediates make the nucleation-controlling temperature higher than the temperature at which the complete target is the most stable cluster.Consequently, complete assembly is typically impossible at a single temperature where nucleation is rate-controlling.Source: ASSEMBLY REQUIRES A TIME-DEPENDENT… · p. 3
  • A significant but surmountable nucleation barrier occurs in a narrow temperature window, while complete-target yield rises only after the barrier disappears.The proposed protocol nucleates at relatively high temperature, then cools further to complete assembly.Source: ASSEMBLY REQUIRES A TIME-DEPENDENT… · p. 3
  • A substantial nucleation barrier slows approach to equilibrium and helps maintain correctly assembled clusters despite off-pathway bonding between partial structures.The authors find that even weak incidental interactions can prevent error-free equilibrium assembly.Source: ASSEMBLY REQUIRES A TIME-DEPENDENT… · p. 4
  • Monte Carlo simulations show that ramp-grown intermediates continue growing on cooling, whereas constant-temperature clusters become arrested; the simulations support the need for a time-dependent protocol.Nucleation first occurs in the predicted window where ΔF‡ ≃ 8 kBT.Source: ASSEMBLY REQUIRES A TIME-DEPENDENT… · p. 4

COORDINATION NUMBER CONTROLS THE NUCLEATION BARRIER

Coordination number shapes nucleation through the topology of the target’s connectivity graph. Low-coordination DNA-brick structures show non-classical behavior, while the close-packed example follows classical nucleation theory.

COORDINATION NUMBER CONTROLS THE NUCLEATION BARRIER

  • At 50% yield, the qc = 4 target coexists with a partial structure missing one cycle, whereas higher-coordination targets have the same free energy as free monomers.Intermediate structures in the higher-coordination examples are globally unstable at all temperatures, consistent with classical nucleation theory.Source: COORDINATION NUMBER CONTROLS THE… · p. 4
  • The qc = 12 structure grows by adding faces to an expanding cuboid, and its free-energy profile near the barrier fits classical nucleation theory.The greater diversity of fragments with the same vertex count smooths the profile near the barrier top.Source: COORDINATION NUMBER CONTROLS THE… · p. 5
  • Low-coordination DNA bricks have a surmountable nucleation barrier at low supersaturation that is relatively insensitive to target size, supporting reliable assembly of large structures.The authors attribute this behavior to the small number of designed connections each brick can make.Source: COORDINATION NUMBER CONTROLS THE… · p. 5
  • The number of vertices needed to complete each additional cycle is a key determinant of nucleation behavior and is controlled by maximum coordination number.The authors caution that higher-coordination structures may require high supersaturation, risking kinetic trapping.Source: COORDINATION NUMBER CONTROLS THE… · p. 5

HETEROGENEOUS BOND ENERGIES IMPROVE KINETICS

The paper finds that heterogeneous bond energies improve DNA-brick nucleation kinetics, contrary to the expectation that equal bond energies would help. In a 74-brick example, random sequences support face-by-face assembly, while monodisperse energies create a larger barrier and competing nuclei.

HETEROGENEOUS BOND ENERGIES IMPROVE KINETICS

  • Broad bond-energy distributions improve nucleation kinetics, whereas minimizing bond-energy variance makes DNA-brick assembly worse.This is consistent with the successful use of random DNA sequences in the original experiments.Source: HETEROGENEOUS BOND ENERGIES IMPROVE… · pp. 5–6
  • For the 74-brick non-convex structure, random sequences support roughly face-by-face assembly, with completion of the third face as the relevant nucleation barrier.The theoretical prediction is confirmed with Monte Carlo simulations.Source: HETEROGENEOUS BOND ENERGIES IMPROVE… · p. 5
  • With monodisperse energies at equivalent mean interaction strength, a larger barrier appears before the first face, and stronger bonds instead lead to kinetic trapping and arrested growth.Multiple competing nuclei appear, and the largest cluster remains poorly configured for further assembly.Source: HETEROGENEOUS BOND ENERGIES IMPROVE… · p. 5
  • A broad distribution selectively stabilizes small, floppy intermediates, reducing barriers because fragments near the barrier top contain fewer cycles.The paper attributes this statistical effect to the low likelihood that random sequences place the strongest bonds in a compact fragment in a large structure.Source: HETEROGENEOUS BOND ENERGIES IMPROVE… · p. 6

DESIGN PRINCIPLES FOR (DNA) NANOSTRUCTURES

The paper links DNA-brick assembly behavior to coordination, bond-energy distributions, and temperature-dependent protocols. It proposes using these principles to tune nucleation barriers and annealing for particular target structures.

DESIGN PRINCIPLES FOR (DNA) NANOSTRUCTURES

  • Low coordination enables slow, controlled nucleation at low supersaturation, while an appreciable barrier within a narrow temperature window limits monomer depletion in one-pot assembly.Random complementary DNA sequences can enhance nucleation, making larger intermediates kinetically accessible at higher temperatures.Source: DESIGN PRINCIPLES FOR (DNA)… · p. 6
  • Slow cooling is an essential non-equilibrium protocol: fast growth forms a stable on-pathway intermediate that must be annealed at lower temperatures to complete the target.Slow diffusion of large intermediates helps keep assembly out of equilibrium and avoid aggregation of partial structures.Source: DESIGN PRINCIPLES FOR (DNA)… · p. 6
  • For a given target, adjusting the statistical distribution of bond energies can tune the nucleation barrier, and complementary sequences can be assigned to achieve that distribution.The approach is presented as a way to improve DNA-brick design beyond randomly selected, uniformly distributed sequences.Source: DESIGN PRINCIPLES FOR (DNA)… · p. 6
  • Understanding a structure's nucleation barrier can guide annealing-protocol optimization, while local coordination changes from fusing adjacent strands can be investigated systematically.The theory is sensitive to target structure and is offered as a means to engineer assembly properties at the design stage; software is provided for protocol design.Source: DESIGN PRINCIPLES FOR (DNA)… · p. 6

METHODS

The methods estimate DNA hybridization free energies and compare theoretical calculations with lattice Monte Carlo simulations. The calculations average properties over sampled sequence sets, while simulations use matched concentration and coordination parameters.

DNA hybridization free energies

  • DNA hybridization free energies are calculated for complementary 8-nucleotide sequences using empirical formulae at specified sodium and magnesium salt concentrations.Calculations average equilibrium yields and free-energy profiles over 1,000 randomly chosen complete sequence sets.Source: DNA hybridization free energies · p. 6

Lattice Monte Carlo simulations

  • Constant-temperature lattice simulations use virtual move Monte Carlo with rigid tetrahedral-patch particles confined to a cubic lattice.A single copy of each required subunit is included in a 62 × 62 × 62 lattice box.Source: Lattice Monte Carlo simulations · pp. 6–7
  • For comparison with simulations, theory uses the same dimensionless concentration, ρ = 62−3, lattice coordination number, qc = 4, and fixed number of dihedral angles, qd = 3.The simulations use those same concentration and structural settings.Source: Lattice Monte Carlo simulations · p. 7

S1. THEORETICAL FREE-ENERGY AND EQUILIBRIUM YIELD CALCULATIONS

The theoretical method represents a target as a connectivity graph and evaluates fragment thermodynamics to predict nucleation-related properties. It assumes clusters form through designed bonds and uses fragment fugacities to characterize yield and structure.

S1. THEORETICAL FREE-ENERGY AND EQUILIBRIUM YIELD CALCULATIONS

  • The target's designed bonds define a connectivity graph from which the method determines thermodynamic properties of intermediate and target structures under a near-equilibrium protocol.The paper points to Ref. 10 for a thorough explanation of the method.Source: S1. THEORETICAL FREE-ENERGY AND… · p. 8
  • Connected target subgraphs are grouped by edge and vertex counts, and the dimensionless grand potential is expressed as a sum over fragment sets assuming clusters form only through designed bonds.The average fugacity of each set also depends on fragment topology, subunit geometry, and solution conditions; the approximation ignores excluded-volume interactions.Source: S1. THEORETICAL FREE-ENERGY AND… · p. 8
  • The fragment model incorporates monomer rotational entropy, unconstrained-dimer dihedral entropy, and solution concentration when estimating fragment fugacities.The bond-energy treatment averages over fragments within a set and samples independently chosen complete sequence sets for DNA-brick structures.Source: S1. THEORETICAL FREE-ENERGY AND… · p. 8
  • The nucleation free-energy definition treats equally sized clusters with varying compositions as indistinguishable because any monomer subset can serve as a nucleation site.This makes the definition appropriate for studying nucleation.Source: S1. THEORETICAL FREE-ENERGY AND… · p. 8
  • Equilibrium yield is defined as the fraction of all solution clusters that are correctly formed, with fragment and target fugacities used in the related calculation.The target fugacity can be replaced by a sum over fragments enforcing correct geometry for structures with higher coordination numbers.Source: S1. THEORETICAL FREE-ENERGY AND… · p. 8

S2. CLASSICAL NUCLEATION THEORY

Classical nucleation theory describes a free-energy barrier for forming an ordered nucleus from an unstable fluid. Its predicted barrier depends on supersaturation and interfacial cost.

S2. CLASSICAL NUCLEATION THEORY

  • Classical nucleation theory predicts a free-energy barrier for forming an ordered structure from an unstable fluid phase.The theory assumes spherical nuclei and considers a nucleus containing V monomers.Source: S2. CLASSICAL NUCLEATION THEORY · p. 8
  • The barrier height and critical-nucleus size vary with the fluid phase’s degree of supersaturation.Source: S2. CLASSICAL NUCLEATION THEORY · p. 8
  • The model’s free-energy terms include the bulk free-energy difference per particle, interfacial cost per area, and ordered-phase number density.Source: S2. CLASSICAL NUCLEATION THEORY · p. 8

S3. CONCENTRATION DEPENDENCE

The calculations use a dimensionless concentration of 62^-3 to compare with lattice Monte Carlo simulations. Changing concentration shifts equilibrium yield and nucleation barrier linearly with ln ρ.

S3. CONCENTRATION DEPENDENCE

S4. ASSEMBLY OF TWO-DIMENSIONAL STRUCTURES

Two-dimensional free-energy profiles depend strongly on coordination number. Compared structures show distinct assembly behavior, and lower-coordinated structures may assemble more robustly.

S4. ASSEMBLY OF TWO-DIMENSIONAL STRUCTURES

  • At 50% yield, similarly sized two-dimensional structures with qc = 3 and qc = 4 exhibit different free-energy profiles; qc = 3 resembles four-coordinated 3D structures, while qc = 4 assembles face-by-face.Source: S4. ASSEMBLY OF TWO-DIMENSIONAL… · p. 9
  • The qc = 3 profile resembles four-coordinated three-dimensional structures, whereas qc = 4 exhibits face-by-face assembly like octahedrally coordinated three-dimensional structures.Source: S4. ASSEMBLY OF TWO-DIMENSIONAL… · p. 9
  • At 50% yield, the qc = 4 structure coexists with fragments having the same monomer count but fewer bonds.Source: S4. ASSEMBLY OF TWO-DIMENSIONAL… · p. 9
  • Two-dimensional structures generally have lower nucleation barriers than similarly sized three-dimensional structures, so lower supersaturation makes targets kinetically accessible and reduces kinetic trapping.Source: S4. ASSEMBLY OF TWO-DIMENSIONAL… · p. 9
  • Calculations suggest lower-coordinated two-dimensional structures, such as the hexagonal lattice, might assemble more robustly in experiments.Source: S4. ASSEMBLY OF TWO-DIMENSIONAL… · p. 9

S5. HYBRIDIZATION FREE-ENERGY DISTRIBUTIONS

The paper compares DNA hybridization free-energy distributions for randomly chosen sequences and sequences selected for monodisperse bond energies. Incidental interactions are approximated to match the lattice Monte Carlo simulations.

S5. HYBRIDIZATION FREE-ENERGY DISTRIBUTIONS

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