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http://openwetware.org/index.php?title=Biomod/2011/TUM/TNT/Project/Structure&action=edit

The structure

Weg zum U

Kriterien für die Struktur

caDNAno

Deformation theory

arm-twist Theorie

[math]\displaystyle{ \vec{r} = \binom{0}{\frac{D}{2}} + R \binom{-sin \alpha}{-cos \alpha} }[/math]

[math]\displaystyle{ \vec{d} = 2 \vec{r} }[/math]

[math]\displaystyle{ d =2 \cdot \sqrt{R^2 + \left( \frac{D}{2} \right)^2 + -RD cos \alpha} }[/math]

arm twist in TEM pictures

In TEM images a dark line can be observed between the two six helix buldles of each arm. If the arms are twisted, this line is no longer exactly in the middle of one arm but "moves around" it. There should also be a variation in the visualized width of the arms due to the asymetric cross-sectional area.

base-twist

To transform the measured arm-twist into a deformation of the structure's base one can assume "the U" as tree cylinders of the same radius. Two of length L (the arms) and one of length B (the middle part of the base). To describe the deformation another cylinder of radius R = radius of the base-cylinder + radius of one arm cylinder and length B is constructed in a way that the center-lines of both arm cylinders are located on its mantle. The torsion of this cylinder (shown in red on the image above) can be directly related to the arm twist.

[math]\displaystyle{ \alpha }[/math] describes the torsion and is divided into [math]\displaystyle{ \alpha = \beta + \gamma }[/math] to include the different projections of the structure on the grid within the theory. The angle [math]\displaystyle{ 2\delta }[/math] between the arms is related to an angle [math]\displaystyle{ \phi }[/math] due to an easier measurement (tree characteristic points are defined as shown in the following picture, one in the middle of the end of the base and one in the middle of the end of each arm)

[math]\displaystyle{ ( L- x ) sin \delta = L sin \frac{\phi}{2} }[/math]

[math]\displaystyle{ x = B \frac{sin \beta}{sin \beta + sin ( \alpha - \beta )} }[/math]

The correlation between [math]\displaystyle{ \delta }[/math] and [math]\displaystyle{ \alpha }[/math] including the different projections described by [math]\displaystyle{ \beta }[/math] is:

[math]\displaystyle{ sin \beta + sin(\alpha - \beta) = \frac{B}{R} sin \delta }[/math]

[math]\displaystyle{ L ( sin \beta + sin (\alpha - \beta )) - B sin \beta = \frac{B L}{R} sin \frac{\phi}{2} }[/math]

The following equation is the arithmetic average of [math]\displaystyle{ L ( sin \beta + sin (\alpha - \beta )) - B sin \beta }[/math] for [math]\displaystyle{ 0 \lt \beta \lt \alpha }[/math]:

[math]\displaystyle{ \frac{1}{\alpha} \int_0^{\alpha} L ( sin \beta + sin (\alpha - \beta )) B\beta = \frac{( cos \alpha -1 ) ( B - 2 L )}{\alpha} }[/math]

[math]\displaystyle{ \frac{cos \alpha -1}{\alpha} = \frac{B L}{R (B - 2 L)} sin \frac{\phi}{2} }[/math]

This equation relates [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \phi }[/math] for an average projection.

This graph shows the result plotted for radius R=6nm, base length B=35nm and total length L=98nm. For this calculation [math]\displaystyle{ \phi }[/math] was calculated for [math]\displaystyle{ \alpha }[/math] from 0 to 133° because of the maximum of [math]\displaystyle{ \epsilon }[/math] at this angle. It can be shown in TEM images that the maximum torsion is below this maximum angle [math]\displaystyle{ \alpha }[/math] by observing the change of the base width. The maximum change confers to a torsion of 180° and can be observed easily, all examined structures show no big change of base width.

base twist in TEM pictures

The twist of the base can be observed at structures that are shown from the side. The twist of the base (in this case the middle part of the base) can be related to a speading of the arms. The following picture shows a structure in a control measurement of BM2 without DNA binders on the left and a twisted structure of the reference BM24 that is designed as a twisted structure on the right.

Design von Referenzstruktur mit Twist

cando

comparison of model structures

TEM images are useful to examine the twist of the arms and in the base.

Experimental considerations

Basic FRET-theory

Certain combinations of dyes exhibit a phenomenon called Förster Resonance Energy Transfer (FRET) when in close proximity. For this to happen, their spectra must match so that the emission of the one dye can excite the other. In short, this means that if the dye with shorter excitation wavelength is excited, it can transfer its energy onto the other dye with the longer excitation wavelength in a radiationless fashion resulting in a shift of emission to longer wavelengths. The extent to which it happens is called FRET efficiency E_FRETand is a sharply decreasing function of the distance between the dyes. The distance d (as derived from the arm twist theory above) where E_FRET is exactly 0.5 is defined as the Förster radius r_F. The following equation describes E_FRET as a function of the angle between the twisted arms:

[math]\displaystyle{ E_{FRET} = \frac{1}{1 + \left( \frac{d}{r_F} \right) ^6} = \frac{{r_F}^6}{{r_F}^6 + (4 \cdot R^2 + D^2 - 4 \cdot R \cdot D \cdot cos \alpha)^3} }[/math] For d=r_F=6,5 nm (see below), D=13 nm and R=4 nm (both parameters derived from the U's structure), α=23,5°. Knowledge of this value is somewhat important for estimating the right experimental conditions to measure within the linear FRET range.

FRET labels

As FRET labels we use the fluorophores Atto 550 and Atto 647N. The Förster distance for this pair is 6.5nm according to AttoTec. Both dyes are commercially available linked to ddNTPs, so they can be attached to oligonucleotides using terminal transferase. The fluorophores not only exhibit a high stability against photobleaching, but also have excitation and fluorescence spectra that fit to the set-up of the self-made fluorescence microscope in our lab. Thus we have not only the possibility to measure FRET at the photospectrometer and the more sensitive RT-PCR, but can also perform single molecule experiments at our TIRF microscope.

FRET-pair positions

Since our U is a 3D object, there are many different options for positioning the fluorophores.
First, they can have different positions in the X-Y-plane, each referring to a particular helix the fluorophores are attached to. We considered a total of 4 symmetric and 3 asymmetric solutions as seen in the figure to the right. The following positions are at the arms' interface: A1 (helix 8), B2 (helix 4), C2 (helix 5), D1 (helix 7) on one arm and A2 (helix 23), B1 (helix 29), C1 (helix 20), D2 (helix 22) on the other. The four symmetric solutions are: A1→A2 and B1→B2 with a distance of 12 nm as well as C1→C2 and D1→D2 with a distance of 5 nm. For our experiments we chose the symmetric solutions A1→A2, B1→B2 and C1→C2, because they complement each other and are more straightforward to analyze due to their symmetry. The expected twist of the arms as seen in the simulation of the naturally (-) twisted positive control is counterclockwise when seen from above. So the pairs B1→B2 and C1→C2 move towards each other with increasing twist until they eclipse, while A1→A2 move apart. For a substance which causes a (+) twist thus deforming the structure clockwise, the opposite pattern could be observed.
Second, different positions along the Z-axis are possible. The relevance of different Z-positions lies in the fact that the fixed basis of the U causes the arms' twist to increase with increasing distance from the base. This way we can adjust the mean displacement due to small molecule binding so it always lies within the linear FRET range. In general, the honeycomb lattice used for the U's construction allows for FRET pairs positioned every 21 basepairs, which is visualized in the figure below. Symmetric solutions align without X-axis shift, whereas asymmetric solutions would expose a 7 basepair shift.

At last, some strategies for attaching the fluorophores to the structure deserved some consideration. Shortening the respective staple to accomodate the labeled nucleotide has the advantage of a well-defined length of the staple. But it is also the less flexible solution because changing the fluorophore's position is not straightforward. On the other hand, extending the existing staple with the labelled nucleotide has the opposite merit profile. Flexibility was more important to us, so we chose to extend the existing staples for labeling.

Survey of folded structures

For a detailed list of all structures, please consult the labbook entry.
One of the most important structures for this project was BM2, which is a simple U shaped origami as described above. Most of the TEM analyses concentrated on this structure. Other important structures were BM12, BM13 and BM14, which contain FRET labels for twist measurements at the three positions mentioned earlier. Each of these structures additionally contained several adapter staples at the bottom, which could be used for immobilization via biotin / neutravidin. Thus, these structures are suited for single molecule fluorescence microscopic examination. BM21 is designed very similar, but with FRET labels positioned for length measurement instead of twist. Finally, BM24 (unlabeled) and BM25 (labels for twist analysis) are intrinsically twisted reference structures.

@@ Line 119: / Line 119: @@
 Second, different positions along the Z-axis are possible. The relevance of different Z-positions lies in the fact that the fixed basis of the U causes the arms' twist to increase with increasing distance from the base. This way we can adjust the mean displacement due to small molecule binding so it always lies within the linear FRET range.  In general, the honeycomb lattice used for the U's construction allows for FRET pairs positioned every 21 basepairs, which is visualized in the figure below. Symmetric solutions align without X-axis shift, whereas asymmetric solutions would expose a 7 basepair shift. <br>
 [[Image:Fluorphore positions cadnano.png|center|x400px]]
-At last, some strategies for attaching the fluorophores to the structure deserved some consideration. Shortening the respective staple to accomodate the labelled nucleotide has the advantage of a well-defined length of the staple. But it also the less flexible solution because changing the fluorophore's position is not straightforward. On the other hand, extending the existing staple with the labelled nucleotide has the opposite merit profile. Flexibility was more important to us, so we chose to extend the existing staples for labelling.
+At last, some strategies for attaching the fluorophores to the structure deserved some consideration. Shortening the respective staple to accomodate the labeled nucleotide has the advantage of a well-defined length of the staple. But it is also the less flexible solution because changing the fluorophore's position is not straightforward. On the other hand, extending the existing staple with the labelled nucleotide has the opposite merit profile. Flexibility was more important to us, so we chose to extend the existing staples for labeling.

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