Figure 1.
Self-assembly model for a 4-component square grid (“2×2”) complex.
The square, rigid components have specific binding rules on each edge denoted by edge colors. Like colored edges interact, whereas edges with different colors and black edges do not interact. An initial, fixed number of components is depleted during self-assembly. At the end of the process, the solution contains a mixture of components, intermediates and complexes.
Figure 2.
Thermodynamic equilibrium is a good predictor of yield for isothermal assembly after long assembly times for 1-dimensional complexes, but not 2- or 3-dimensional complexes.
Assembly yields for a (a) 1×5 line complex, (b) 2×2, (c) 3×3, (d) 4×4 and (e) 5×5 square grid complex and (f) 2×2x2 cube complex as a function of the dimensionless temperature parameter, . Inset diagram depicts the complex. Numbers on the components in the complex indicate component identity (e.g. component “1” is different than component “2”). The dashed line indicates thermodynamic equilibrium. Dimensionless reaction time is defined as
where
is the macroscopic forward reaction rate constant and
is the initial concentration of components. Colored bars and boxes below figures represent the four different assembly regimes (Text S3). The assembly funnel regime is considered to be where the complex is thermodynamically favored (i.e.,
) and assembly is rapid such that
. Assembly “snapshots” (below graphs) are taken at
and
(top row),
,
, and
(bottom row) and comprised of ten random species drawn from the reaction mixture, weighted by concentration (Text S4). Error bars indicate the standard deviation of the reported quantity after 10 simulations and where omitted, are <1%. Here and elsewhere unless otherwise noted, there is no bond coupling (
).
Figure 3.
An assembly funnel means that complex assembly occurs via a small number of pathways.
The possible set of reaction pathways govern assembly outcome under rearrangement-limited conditions, whereas thermodynamically favorable pathways govern assembly outcome in the assembly funnel regime. (a) Conformational entropy () of the system under different assembly conditions as a function of assembly time,
. (b) Reference energy distributions of a 3×3 square grid complex based on thermodynamics and assembly configuration. Color spectrum indicates the number of bonds in an assembly. (c) Partition of energies at different times during self-assembly in the assembly funnel regime at
(green box), rearrangement-limited conditions at
(blue box), and during an anneal (black box). Over the course of an anneal,
transitions from -6 to 6, spending
at 100 different linearly decreasing isothermal conditions. Values at the top right are complex yields. Inset plots show detail. Error bars <1%.
Figure 4.
Complexes form rapidly in the assembly funnel regime.
Yield of 3×3 square grid complex as a function of reaction time by assembling via annealing and at various isothermal assembly conditions: (orange, nucleation-limited),
(green, assembly funnel)
(blue, parallel assembly pathways and rearrangement-limited). Inset plot (top left) depicts yield during an anneal as a function of interaction strength for different reaction times:
(salmon),
(beige), and
(purple). Inset diagram (bottom right) depicts the complex.
Figure 5.
The amount of bond coupling, or additivity of bond energies during cooperative binding steps does not significantly affect assembly yields above a small threshold.
Yield of a 3×3 square grid complex as a function of the bond coupling constant, under many isothermal assembly conditions (solid lines, color) and after an anneal (black) for reaction time
. Dashed lines show yields at thermodynamic equilibrium for isothermal conditions with the same color. Error bars <1%.
Figure 6.
Design of components so that particular assembly pathways are favored can drastically increase assembly yields.
(a) Schematic of spiral complex assembly via the favored assembly pathway. On the favored assembly pathway, assembly begins with the “L” shaped component, labeled “1”. At each assembly step, a component attaches through two interfaces (following the green arrow). Other components can only attach through one. Lengths of reaction arrows indicate propensities in the assembly funnel regime. Assembly yields for a (b) 2×2 (4 component), (c) 3×3 (9 component) and (d) 4×4 (16 component) spiral complex as a function of a dimensionless temperature parameter, . Inset diagram depicts the complex and numbers on the components in the complex indicate component identity. Colored bars below the figure represent the four different assembly regimes for spiral complexes and grid complexes containing the same number of components. Error bars <1%.