# Spherical Container

## Overview

With our sphere design, we were able to:

• Fold and characterize through atomic force microscopy (AFM) the original Han et al. sphere, which we call the "closed" sphere.
• Fold and characterize through AFM our "open" sphere in which we removed all equator staple strands.
• Test various lock mechanisms to transition between the closed and open states of the sphere.
• Analyze the scaffold-staple-lock system to find out why certain lock designs were unsuccessful at closing the sphere.

Note that due to the sphere being a single-layer DNA structure and a popular shape for water droplets, it was hard to characterize via transmission electron microscopy.

## Folding Origami

### Gel

 At right we have a 2% agarose gel of spheres in the closed and open states. The circled bands indicate possible locations of desired origami in the gel. We expect the closed and open state spheres to run faster than scaffold, as they are more compact. Indeed, closed spheres can be found in the band boxed in the "Closed Sphere" lane and open spheres can be found in the middle box of the "Open Sphere" lane. AFM imaging of the top band proved inconclusive, but we theorize that the higher band is an aggregated form of the open sphere, perhaps two open spheres bound together.

### Closed Sphere

 We successfully recreated the Han et al. sphere down to the base, and were able to characterize the structure under AFM. The AFM image at right shows an unpurified sample of closed sphere. We find circles consistently around 50 nm in diameter. Han et al. designed the closed sphere to be 42 nm in diameter; considering flattening that may occur when the sphere is AFMed and the width of the AFM cantilever, these circles are what we expect closed spheres to look like under AFM.

## Opening the Sphere

### Open Sphere

 We removed the equator staples that hold the two halves of Han's sphere together to create an open state held together only by a scaffold crossover. To make this open state close-able, we designed equator staples with extensions that can bind to a "lock" strand that closes the sphere. To the right is an AFM image of gel-purified open spheres with equator staples compatible with such a closing mechanism. We see that each structure is comprised of two circles of diameter ~50 nm, which is what we expect to see of open spheres when their two halves are splayed open along the plane of the AFM image.
 Here is a zoomed-in version of open spheres. In this image we see rings in addition to circles, indicating an orientation of open spheres where the edges of each hemisphere point up (imagine the AFM image that two bowls side by side on a table would create).

## Closing the Sphere

We tested a variety of locks to close the open state of the sphere, changing lock length, composition, and concentration relative to open spheres. Unfortunately, the AFM images we obtained all indicated that these spheres were still in an open state.

### Troubleshooting

 The first lock we tried was 43 bp long, of sequence 5'-CGTGATTGTTACCAG-/iSpPC/-TTCGTTAGTTCAGACAG-CAGACAGACAG-3', where the bolded portion indicates the toehold for a strand displacement opening mechanism and /iSpPC/ indicates a photo-cleavable spacer for a photo-cleavage opening mechanism. Note that 15 bp of the lock is complementary to one equator staple, and 18 bp is complementary to another. When this lock did not close the spheres, we thought of a few reasons why this might be the case. Our thoughts are summarized in the diagram to the right. In the diagram, Scenario (1) is the desired case. Scenario (2) occurs when a lock binds to each of the equator staple extensions. Since the complementary region is 15 or 18 bp long, once bound the lock strand will not dislodge. Thus, if the concentration of locks is high enough, equator staples will be saturated with locks and the two halves of the open sphere will never be brought together by a single lock. Scenario (3) occurs when one lock binds to two different open spheres. Again, since the complementary regions between lock and staples are long, the lock will not dislodge.

We also explored the kinetics of scaffold, staple, and lock interaction for this system to see whether Scaffold:Staple complexes or Staple:Lock complexes form first.

Calculations of which interaction, Scaffold:Staple or Staple:Lock, will ultimately win out:
There are 100 nM of any individual staple and 10 nM of scaffold in solution. Assume the concentration of lock is 900 nM; because there are 9 locks per sphere, this actually means the lock is the same concentration as an individual staple. The reaction that proceeds is:
Scaffold + Staple --> Scaffold:Staple
At Tm1, the melting temperature of the Scaffold:Staple complex, there are 5 nM of Scaffold:Staple, 5 nM of Scaffold, and 95 nM of Staple.
$K_{eq}=\frac{[Scaffold:Staple]}{[Scaffold][Staple]} = \frac{5 nM}{5 nM*95 nM} = \frac{1}{95 nM} = 10^7 / M$
There are 100 nM of any individual staple complementary to the lock and 900 nM of lock in solution. The reaction that proceeds is:
Staple + Lock --> Staple:Lock
At Tm2, the melting temperature of the Staple:Lock complex, there are 50 nM of Staple:Lock, 50 nM of Staple, and 850 nM of Lock.
$K_{eq}=\frac{[Staple:Lock]}{[Staple][Lock]} = \frac{50 nM}{50 nM*850 nM} = \frac{1}{850 nM} = 10^6 / M$
At melting temperature Tm,
$\Delta G^{\circ}=-R T_m \log{K_{eq}}=\Delta H-T_m \Delta S$
$T_m=\frac{\Delta H^{\circ}}{\Delta S^{\circ}-R \log{K_{eq}}}$
Assuming that ΔH° and ΔS° are approximately the same for Scaffold:Staple and Staple:Lock interactions, a larger (more positive) Keq results in a larger numerator in the expression above. Since ΔS° is negative, a larger Keq results in a more negative Tm.
So, since Keq is larger for the Scaffold:Staple reaction, Tm is lower for the formation of Scaffold:Staple than for Staple:Lock. In other words, when temperature is being lowered in the thermocycler, Staple:Lock forms before Scaffold:Staple.

From these calculations we can conclude that adding a high concentration of lock may not help the open spheres close.

To avoid the problem of locks not dislodging once bound to a staple, we designed shorter locks that have complementary regions of ~8 bp rather than ~16 bp. This shorter complementary sequence allows locks and staples to periodically dissociate, meaning scenarios (2) and (3) in the above diagram are not permanent.

We tested these "8 bp locks" but did not see conclusive evidence of open spheres closing. However, we believe that with additional time it is entirely possible to design a system of locks that will close the open sphere.