How big of a cube can DNA lift?
When you’re first exposed to nanotechnology, it can be difficult to grasp the immense difference in scale that nano-sized objects have. It’s helpful to think through the actual mechanics of how objects that size interact. For example, we use DNA linkers to bind DNP cubes together into useful structures. Just how strong are the bonds between DNA linkers? Imagine you’re playing a nano-sized version of the claw game, except instead of a mechanical claw, you’re going to lift objects with using DNA. It would look something like this:
How large of a DNP cube could you lift?
While there are many ways to bind nanoparticles together (e.g. covalent bonds, electrostatic attraction, hydrophilic/hydrophobic interactions, etc.), we typically use the stickiness of double stranded DNA. We do this by first painting single-stranded DNA on the cube's faces:
When two cubes get close, they will bind together to form double stranded DNA, if and only if they have complementary sequences:
Background. DNA is held together by hydrogen bonds and stacking interactions. At least a couple sources list the binding force of DNA as being in the piconewton (pN) range . Let’s assume the force required to break apart one of the short DNA strands holding the cubes together is 4 pN. The total force holding the cubes together should be proportional to the number of DNA strands connecting them. If you have 2 DNA strands binding the cubes together, it should take 8 pN of force to break them apart. If you have 3 DNA strands, it should take 12 pN. If 10 DNA strands, it should take 40 pN. The more DNA strands you have, the stronger the cubes will bind together.
How many DNA strands can you fit on the face of a cube? Clearly it depends on the size of the cube. Suppose you have a cube of length 10 nanometers (nm). The area of a single cube face would be 10 nm × 10 nm = 100 square nanometers. DNA is 2 nm wide, giving it a cross-sectional area of roughly 2 nm × 2 nm = 4 square nanometers . Assuming you cover the surface of the 10 nm cube completely, we should be able to fit roughly 25 DNA strands onto a single face.
How big is the force that binds cubes together? Since DNA has a force of 4 pN per strand and a 10 nm cube can fit 25 DNA strands on each face, the total force binding the cubes together will be 4 pN per strand × 25 strands = 100 pN. What if we use a larger cube? We can repeat the analysis above to estimate the maximum number of DNA strands that can fit on a cube face is
number of strands ≈ (cube length in nm) × (cube length in nm) / 4
Using this formula, we can see that a 20 nm cube can fit 100 DNA strands, a 100 nm cube can fit 2500 DNA strands, a 1000 nm cube can fit 250,000 DNA strands, etc. If each strand delivers 4 pN of binding force, than the total force holding the cubes together is given by the equation
total force in piconewtons ≈ (cube length in nm) × (cube length in nm)
How much does the cube weigh? In order to lift an object, you need to overcome its weight, or in physics-speak, the gravitational force pulling it down. The force of gravity pulling the cube down is equal to its mass multiplied by the acceleration of gravity, which is roughly 9.8 meters per second squared. We can calculate the cube’s mass by multiplying its density times its volume. The volume is simply the length cubed. We’ll assume the nanocube is made of silver, which has a bulk density of 10.5 grams per cubic centimeter. Combining these facts, we can write the equation
weight in grams = (0.000 000 000 10) × (length in nm) × (length in nm) × (length in nm),
Notice that the particle’s weight is proportional to its length cubed, whereas the binding force is only proportional to the its length squared. This means that as the cube grows in size, the gravitational force pulling it down grows faster than the force of the DNA pulling it up. Eventually, the weight of the cube will be too much, and the DNA strands will not be strong enough to lift it. At that point, the DNA strands will break, and the cube will fall. We can see that on a plot of force versus cube length:
We see that the gravitational force acting on the cube (green) grows faster than the DNA binding force (orange). The forces intersect at a cube length of approximately 2.4 meters. That’s an 8 foot long cube weighing 165 tons! Evidently, DNA is very strong if you have enough of them.
I should point out that for a real cube, the largest size that DNA can lift is almost certainly smaller than the result computed here. Nanocube faces are very flat, whereas bulk silver is rough and bumpy. When you stack bumpy cubes on top of each other, the bumps that stick out will be in contact with the other cube, but the dimples won’t be in contact with the other cube. Since the bumps and dimples will be larger than the short DNA strands, the strands located in the dimples will never reach far enough to bind with the DNA on the other cube.
 A piconewton is 0.000 000 000 001 newtons of force. For example, see Phys.org’s “Measuring forces in the DNA molecule” or PicoTwist’s “Forces involved at the biological level”.
 The cross-sectional area of DNA is more accurately described as a circle, but for an order of magnitude estimation such as this, the difference between the area of a square and the area of a circle will not be significant.
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