Biomod/2011/TUM/TNT/Project

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<h2>Relevance of DNA - Small Molecule Interactions</h2>
<h2>Relevance of DNA - Small Molecule Interactions</h2>
<table cellspacing="0" cellpadding="2">
<table cellspacing="0" cellpadding="2">
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<tr bgcolor="#f5f5f5"><td valign="top">Containing the blueprint for every function and structure of life, DNA plays an essential role in most fields of the biosciences. Because of this reason, detailed knowledge of compounds that interact with DNA is of great importance. For instance, small DNA binding molecules including intercalators and minor/ major groove binders can cause changes in the geometry of double-helical DNA domains. Such changes can affect the dynamics of transcriptional regulation and the activity of nucleases. To name an example, the minor binder Netropsin is a potent inhibitor of bacterial DNA gyrases with a broad range of activity, including the notorious nosocomial pathogen Pseudomonas aeruginosa.<br>
+
<tr bgcolor="#f5f5f5"><td valign="top">Containing the blueprint for every function and structure of life, DNA plays an essential role in most fields of the biosciences. Because of this reason, detailed knowledge of compounds that interact with DNA is of great importance. For instance, small DNA binding molecules including intercalators and minor / major groove binders can cause changes in the geometry of double-helical DNA domains. Such changes can affect the dynamics of transcriptional regulation and the activity of nucleases. To name an example, the minor groove binder Netropsin is a potent inhibitor of bacterial DNA gyrases with a broad range of activity, including the notorious nosocomial pathogen Pseudomonas aeruginosa.<br>
Hence, small DNA binding molecules can represent candidate compounds with antibiotic or tumor-repressing activity.  Elucidating the microscopic structural changes caused by small molecules can thus help to select candidate molecules as well as shed light on functional mechanisms.</td><td>[[Image:DNA Netropsin.png|thumb|200x200px|right|Fig. 1: Crystal structure of the minor groove binder Netropsin bound to DNA.]]</td></tr>
Hence, small DNA binding molecules can represent candidate compounds with antibiotic or tumor-repressing activity.  Elucidating the microscopic structural changes caused by small molecules can thus help to select candidate molecules as well as shed light on functional mechanisms.</td><td>[[Image:DNA Netropsin.png|thumb|200x200px|right|Fig. 1: Crystal structure of the minor groove binder Netropsin bound to DNA.]]</td></tr>
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<h2>Why DNA Origami?</h2>
<h2>Why DNA Origami?</h2>
The working principle of DNA origami is based on the highly specific interaction behavior of DNA. Because of this specificity, the binding behavior of two strands can be encoded in the base sequence of said strands. By including sequences complementary to two strands on another, a trimolecular system can be created where one strand binds two different ones together. By including more and more complementary sequences on a single strand, increasingly complex configurations can be created. DNA Origami uses this modular behavior to encode the shape of the planned structure in the sequences of the used DNA strands. Designing a new structure is generally done according to the following steps. During the computer-aided design of the structure, a long single stranded DNA called the 'scaffold' is first laid out such that it fills out the shape of the planned structure. Next, short DNA strands called 'staples' are chosen such that they are complementary to partial sequences of the scaffold. By binding to those complementary sequences, the staples force the scaffold into a set conformation, removing degrees of freedom. With each bound staple, more and more helices with fixed relative positions are created. Once all bases of the scaffold are hybridized, the scaffold is forced into one specific shape.<br>
The working principle of DNA origami is based on the highly specific interaction behavior of DNA. Because of this specificity, the binding behavior of two strands can be encoded in the base sequence of said strands. By including sequences complementary to two strands on another, a trimolecular system can be created where one strand binds two different ones together. By including more and more complementary sequences on a single strand, increasingly complex configurations can be created. DNA Origami uses this modular behavior to encode the shape of the planned structure in the sequences of the used DNA strands. Designing a new structure is generally done according to the following steps. During the computer-aided design of the structure, a long single stranded DNA called the 'scaffold' is first laid out such that it fills out the shape of the planned structure. Next, short DNA strands called 'staples' are chosen such that they are complementary to partial sequences of the scaffold. By binding to those complementary sequences, the staples force the scaffold into a set conformation, removing degrees of freedom. With each bound staple, more and more helices with fixed relative positions are created. Once all bases of the scaffold are hybridized, the scaffold is forced into one specific shape.<br>
-
If the design of an origami contains certain targeted insertions and deletions of base pairs, it is possible to create a built-in twist or curvature [[http://openwetware.org/wiki/Biomod/2011/TUM/TNT/Extras#References Dietz et. al., 2009]]. This work was inspiring because it implies the general idea of an origami's overall shape to be liable to distortions of it's microstructure. According to this concept, the above mentioned constraints within the DNA Origami allow the effects of multiple DNA binders to be transported along the structure so that they sum up, amplifying otherwise immeasurable influences. Furthermore, the unparalleled positional control of DNA Origami allows us to place probe molecules on the structure with high accuracy, permitting the observation of minute changes.
+
If the design of an origami contains certain targeted insertions and deletions of base pairs, it is possible to create a built-in twist or curvature [[http://openwetware.org/wiki/Biomod/2011/TUM/TNT/Extras#References Dietz et. al., 2009]]. This work was inspiring because it implies the general idea of an origami's overall shape to be liable to distortions of its microstructure. According to this concept, the above mentioned constraints within the DNA Origami allow the effects of multiple DNA binders to be transported along the structure so that they sum up, amplifying otherwise immeasurable influences. Furthermore, the unparalleled positional control of DNA Origami allows us to place probe molecules on the structure with high accuracy, permitting the observation of minute changes.
<h1>Project Specification</h1>
<h1>Project Specification</h1>
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*Provide means for quantifying how the target molecule changes helical pitch and length of double-helical DNA domains.<br>
*Provide means for quantifying how the target molecule changes helical pitch and length of double-helical DNA domains.<br>
We set out to develop a self-assembled DNA origami device that amplifies microscopic structure changes imposed on its constituent double-helical DNA domains by a large global conformational change. The global structure alteration becomes detectable either by direct imaging with transmission electron microscopy (TEM) but ideally also in a simple fluorometric assay with equipment that is available to many laboratories in the world.<br>
We set out to develop a self-assembled DNA origami device that amplifies microscopic structure changes imposed on its constituent double-helical DNA domains by a large global conformational change. The global structure alteration becomes detectable either by direct imaging with transmission electron microscopy (TEM) but ideally also in a simple fluorometric assay with equipment that is available to many laboratories in the world.<br>
-
We have focussed on changes in helical pitch and tested our device with a set of different DNA binders from the three major classes, including <b>spermine (major groove binding), ethidium bromide (intercalating),</b> and <b>DAPI (minor groove binding)</b> in different concentrations by direct imaging with TEM. In accordance to computational models that we made for our device, the local deformations led to observable global structural changes. In the light of the time constraints of this project we therefore consider the goals accomplished by our proof-of-principle experiments with TEM alone.<br>
+
We have focussed on changes in pitch and length and tested our device with a set of different DNA binders from the three major classes, including <b>spermine (major groove binding), ethidium bromide (intercalating),</b> and <b>DAPI (minor groove binding)</b> in different concentrations by direct imaging with TEM. In accordance to computational models that we made for our device, the local deformations led to observable global structural changes. In the light of the time constraints of this project we therefore consider the goals accomplished by our proof-of-principle experiments with TEM alone.<br>
However, since access to TEM microscopes is limited, for practical application of the device it is desirable that it can function in simple fluorometric assays. We have therefore also taken first steps toward using fluorescent signals to report on structural changes that our device undergoes upon binding of small molecules. We present some promising preliminary results obtained with single-molecule FRET microscopy and bulk photometry. 
However, since access to TEM microscopes is limited, for practical application of the device it is desirable that it can function in simple fluorometric assays. We have therefore also taken first steps toward using fluorescent signals to report on structural changes that our device undergoes upon binding of small molecules. We present some promising preliminary results obtained with single-molecule FRET microscopy and bulk photometry. 
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<h3>Breaking the Staples</h3>
<h3>Breaking the Staples</h3>
-
With the completion of the routed scaffold we just have to add complimentary pieces of DNA which fit into the honey comb pattern of theU. But there are several problems which had to be considered: <br>
+
With the completion of the routed scaffold we just have to add complementary pieces of DNA which fit into the honey comb pattern of theU. But there are several problems which had to be considered: <br>
* Staples must not be longer than 50 bases, because of rising inaccuracy of the sequence with rising staples length.
* Staples must not be longer than 50 bases, because of rising inaccuracy of the sequence with rising staples length.
* Staples must not be shorter than 14 bases, because a certain length is needed so that the staple is most of the time bound at room. temperature
* Staples must not be shorter than 14 bases, because a certain length is needed so that the staple is most of the time bound at room. temperature
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<tr bgcolor="#f5f5f5"><td valign="top">Since we also wanted to perform single molecule FRET measurements with our structure, we finally had to add adapter staples to the base of theU. We used biotin labeled staples to bind theU to the surface of a microscope slide.</td><td>[[Image:adaptors.png|right|x200px | thumb | Fig. 5 Adaptors]]</td></tr>
+
<tr bgcolor="#f5f5f5"><td valign="top">Since we also wanted to perform single molecule FRET measurements with our structure, we finally had to add adapter staples to the base of theU. We used biotin labeled staples to bind theU to the surface of a microscope slide.</td><td>[[Image:adaptors.png|right|x200px | thumb | Fig. 5 Adaptors]]</td></tr>
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<tr bgcolor="#f5f5f5"><td valign="top">In order to have a positive control, we designed a structure which has an internally induced twist. This was done by adding an extra base pair every 21 base pairs in every single helix of theU. Since there is a cross over to another helix every 7 base pairs, this additional base pairs induces a local torque which leads to a macroscopic twist in the structure. This conformational shift could later clearly be seen in finite elements computer simulations and TEM images. In fig. 6 the 21 base pair tokens are indicated by the 2 black dashed lines and the additional base pairs by the little loops with '1' inside the scaffold.</td><td>[[Image:loop.png|right|x200px | thumb | Fig. 6 Reference structure. 21 base pair tokens: 2 black dashed lines. Additional base pairs: loops with '1' inside the scaffold.]]</td></tr>
+
<tr bgcolor="#f5f5f5"><td valign="top">In order to have a positive control, we designed a structure which has an internally induced twist. This was done by adding an extra base pair every 21 base pairs in every single helix of theU. Since there is a crossover to another helix every 7 base pairs, this additional base pairs induces a local torque which leads to a macroscopic twist in the structure. This conformational shift could later clearly be seen in finite elements computer simulations and TEM images. In fig. 6 the 21 base pair tokens are indicated by the 2 black dashed lines and the additional base pairs by the little loops with '1' inside the scaffold.</td><td>[[Image:loop.png|right|x200px | thumb | Fig. 6 Reference structure. 21 base pair tokens: 2 black dashed lines. Additional base pairs: loops with '1' inside the scaffold.]]</td></tr>
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Revision as of 20:02, 2 November 2011

TUM NanU - Home





Project


Contents

The Idea

Relevance of DNA - Small Molecule Interactions

Containing the blueprint for every function and structure of life, DNA plays an essential role in most fields of the biosciences. Because of this reason, detailed knowledge of compounds that interact with DNA is of great importance. For instance, small DNA binding molecules including intercalators and minor / major groove binders can cause changes in the geometry of double-helical DNA domains. Such changes can affect the dynamics of transcriptional regulation and the activity of nucleases. To name an example, the minor groove binder Netropsin is a potent inhibitor of bacterial DNA gyrases with a broad range of activity, including the notorious nosocomial pathogen Pseudomonas aeruginosa.
Hence, small DNA binding molecules can represent candidate compounds with antibiotic or tumor-repressing activity.  Elucidating the microscopic structural changes caused by small molecules can thus help to select candidate molecules as well as shed light on functional mechanisms.
Fig. 1: Crystal structure of the minor groove binder Netropsin bound to DNA.
Fig. 1: Crystal structure of the minor groove binder Netropsin bound to DNA.

Why DNA Origami?

The working principle of DNA origami is based on the highly specific interaction behavior of DNA. Because of this specificity, the binding behavior of two strands can be encoded in the base sequence of said strands. By including sequences complementary to two strands on another, a trimolecular system can be created where one strand binds two different ones together. By including more and more complementary sequences on a single strand, increasingly complex configurations can be created. DNA Origami uses this modular behavior to encode the shape of the planned structure in the sequences of the used DNA strands. Designing a new structure is generally done according to the following steps. During the computer-aided design of the structure, a long single stranded DNA called the 'scaffold' is first laid out such that it fills out the shape of the planned structure. Next, short DNA strands called 'staples' are chosen such that they are complementary to partial sequences of the scaffold. By binding to those complementary sequences, the staples force the scaffold into a set conformation, removing degrees of freedom. With each bound staple, more and more helices with fixed relative positions are created. Once all bases of the scaffold are hybridized, the scaffold is forced into one specific shape.
If the design of an origami contains certain targeted insertions and deletions of base pairs, it is possible to create a built-in twist or curvature [Dietz et. al., 2009]. This work was inspiring because it implies the general idea of an origami's overall shape to be liable to distortions of its microstructure. According to this concept, the above mentioned constraints within the DNA Origami allow the effects of multiple DNA binders to be transported along the structure so that they sum up, amplifying otherwise immeasurable influences. Furthermore, the unparalleled positional control of DNA Origami allows us to place probe molecules on the structure with high accuracy, permitting the observation of minute changes.

Project Specification

Our goals were to develop a simple and fast assay for

  • Identifying whether or not a target compound binds DNA in a structure-altering fashion and
  • Provide means for quantifying how the target molecule changes helical pitch and length of double-helical DNA domains.

We set out to develop a self-assembled DNA origami device that amplifies microscopic structure changes imposed on its constituent double-helical DNA domains by a large global conformational change. The global structure alteration becomes detectable either by direct imaging with transmission electron microscopy (TEM) but ideally also in a simple fluorometric assay with equipment that is available to many laboratories in the world.
We have focussed on changes in pitch and length and tested our device with a set of different DNA binders from the three major classes, including spermine (major groove binding), ethidium bromide (intercalating), and DAPI (minor groove binding) in different concentrations by direct imaging with TEM. In accordance to computational models that we made for our device, the local deformations led to observable global structural changes. In the light of the time constraints of this project we therefore consider the goals accomplished by our proof-of-principle experiments with TEM alone.
However, since access to TEM microscopes is limited, for practical application of the device it is desirable that it can function in simple fluorometric assays. We have therefore also taken first steps toward using fluorescent signals to report on structural changes that our device undergoes upon binding of small molecules. We present some promising preliminary results obtained with single-molecule FRET microscopy and bulk photometry. 

The Structure

Criteria for the Structure

Besides the general requirements for a structure as stated in the project idea and specification, there were some needs that should also be fulfilled by a candidate structure.

  • Simple observation by transmission electron microscopy.

In the first place, such structure should allow for a simple observation of the twist by transmission electron microscopy so that changes in twist can easily be recognized.
We considered a U-shaped DNA origami to be favorable, since it has two arms that can twist against each other. Also, if they were designed with a flattened cross section, their twist is more distinguished in TEM images.

  • Placement for FRET dyes.

Most important, they allow for the placement of FRET dyes on different positions to measure their relative movement. Because of this, dimensions lie within fixed borders so that the linear FRET range can be fully exploited. Long arms are best suited for a good enhancement of the small structural changes imposed on DNA by the binding of a single DNA binder and for placing FRET dyes along their axes to measure elongation.

  • Flexibility

Another important factor was flexibility, since we wanted to measure deformations.

  • Folding

Folding should be proper. Therefore and to obtain a good yield we decided to build a monomeric structure.


We considered the U-shape to be an elegant solution to the general design criteria and also the particular needs to be met.

caDNAno

Fig. 2 Cross section. Orange circles: helices of the two arms. Grey circles: helices of the base.
Fig. 2 Cross section. Orange circles: helices of the two arms. Grey circles: helices of the base.

The design of the three dimensional shape was done with caDNAno (a free program for easily designing 3D DNA origamis; see http://cadnano.org/ for further information). So we created a suited cross section, routed the scaffold's path through the cross section, broke the staples and added final modifications like adapter staples for the immobilization on glass slides.

Cross Section

theU consists in principle of two 10 helix bundles which are connected on the broad side with a third 10 helix bundle. So we have two long deformable arms connected by a base. The cross section can be seen in fig. 2. The orange circles are the helices of the two arms and the 10 grey ones the additional helices of the base. The long arms are now able to amplify the deformation of the single helices in our structure.

Scaffold Routing

Fig. 3 Scaffold path
Fig. 3 Scaffold path
Based on the cross section, we could begin routing the circular scaffold through the pattern of helices. The scaffold first goes through one 10 helix bundle and then changes over to the next 10 helix bundle as one can see in fig. 3 (the red arrows indicate the direction of the scaffold path). It is commonly known that this enables easy folding of the structure.

(refer to our results from folding and purification)


Breaking the Staples

With the completion of the routed scaffold we just have to add complementary pieces of DNA which fit into the honey comb pattern of theU. But there are several problems which had to be considered:

  • Staples must not be longer than 50 bases, because of rising inaccuracy of the sequence with rising staples length.
  • Staples must not be shorter than 14 bases, because a certain length is needed so that the staple is most of the time bound at room. temperature
  • Prefer donating half a cross over of a staple instead of breaking a seven base long bound part of a staple.

Following these rules we have almost finished the final layout of theU.

The U v02 18.json


The grey staples belong to the core of our structure and are needed to build the basic shape. The staples marked in red can be added to the folding reaction if uncontrolled aggregation arises (if included the single stranded poly-T domains build random coils which suppress the base stacking interaction with another structure). In our case basically no aggregation occurred and we did not need to add the extra poly-T staples.
Fig. 4 caDNAno design. Grey staples: build the basic shape. Red staples: if uncontrolled aggregation arises.
Fig. 4 caDNAno design. Grey staples: build the basic shape. Red staples: if uncontrolled aggregation arises.

Modifications

Since we also wanted to perform single molecule FRET measurements with our structure, we finally had to add adapter staples to the base of theU. We used biotin labeled staples to bind theU to the surface of a microscope slide.
Fig. 5 Adaptors
Fig. 5 Adaptors


Design of Reference Structure with Twist

In order to have a positive control, we designed a structure which has an internally induced twist. This was done by adding an extra base pair every 21 base pairs in every single helix of theU. Since there is a crossover to another helix every 7 base pairs, this additional base pairs induces a local torque which leads to a macroscopic twist in the structure. This conformational shift could later clearly be seen in finite elements computer simulations and TEM images. In fig. 6 the 21 base pair tokens are indicated by the 2 black dashed lines and the additional base pairs by the little loops with '1' inside the scaffold.
Fig. 6 Reference structure. 21 base pair tokens: 2 black dashed lines. Additional base pairs: loops with '1' inside the scaffold.
Fig. 6 Reference structure. 21 base pair tokens: 2 black dashed lines. Additional base pairs: loops with '1' inside the scaffold.

The U v02 biotinadapter 21++.json


Cando

In addition to our structural considerations we also did computer simulations on our structures. They were performed with Cando (a finite elements simulation based on our *.json files and mechanical properties of dsDNA). Figures 7a and b show the simulation of the nontwisted structure from two views. In comparison figures 8 a and b show the pre-twisted structure from the same projections. One can clearly see the twist and spread of the arms in fig. 8 a,b. These simulations were used for further theoretical and experimental considerations.

Fig. 7a cando simulation of the structure BM2 view 1.
Fig. 7a cando simulation of the structure BM2 view 1.
Fig. 7b  cando simulation of the structure BM2 view 2.
Fig. 7b cando simulation of the structure BM2 view 2.
Fig. 8a cando simulation of the twisted control BM21 view 1.
Fig. 8a cando simulation of the twisted control BM21 view 1.
Fig. 8b  cando simulation of the twisted control BM21 view 2.
Fig. 8b cando simulation of the twisted control BM21 view 2.


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. 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.


Theoretical Background

Twist of the Arms

Theoretical considerations

Fig. 9 Cylindric model for the twist of the arms.
Fig. 9 Cylindric model for the twist of the arms.

The origin of z is located at the bottoms of the arms where the base ends as depicted in the figure 9 to the left. The angle α of the torsion of the arms shows a linear dependency on z α0 defines a position in the undeformed state, Δα + α0 is the maximum torsion at the end of the arms, B is the length of the base and L is the total length of the structure):


\alpha (z) = \frac{z}{L-B} \cdot  \Delta\alpha + \alpha_0

The distance between two points on the mantle of the cylinders that describe the arms can be easily derived by vector analysis and results in:


d = 2 \cdot \sqrt{R^2 + \left( \frac{D}{2} \right)^2 - R \cdot D \cdot cos \left( \alpha (z) \right)}

The following diagram (fig. 10) shows a relation between Δα and d for R=4nm, D=13nm and α0=0°.

Fig. 10 Distance between two points on the cylinder mantles representing the arms over torsion angle Δα.
Fig. 10 Distance between two points on the cylinder mantles representing the arms over torsion angle Δα.


Observing 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.

Fig. 11a Top view of BM2 without DNA-binders.
Fig. 11a Top view of BM2 without DNA-binders.
Fig. 11b Top view of BM2 with approximately one EtBr molecule every 7bp.
Fig. 11b Top view of BM2 with approximately one EtBr molecule every 7bp.

Twist of the Base

Theoretical Considerations

Fig. 12 Cylindric model for the twist of the base.
Fig. 12 Cylindric model for the twist of the base.

To transform the measured arm-twist into a deformation of the structure's base one can assume theU as three 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.

α describes the torsion and is divided into α = β + γ to include the different projections of the structure on the grid within the theory. The angle between the arms is related to an angle φ due to an easier measurement (tree characteristic points are defined as shown in the following figure 13, one in the middle of the end of the base and one in the middle of the end of each arm).

Fig. 13 Representative TEM image including the way φ is measured.
Fig. 13 Representative TEM image including the way φ is measured.


( L- x ) sin \delta = L sin \frac{\phi}{2}


x = B \frac{sin \beta}{sin \beta + sin ( \alpha - \beta )}


The correlation between δ and α including the different projections described by β is:


sin \beta + sin(\alpha - \beta) = \frac{B}{R} sin \delta


L ( sin \beta + sin (\alpha - \beta )) - B sin \beta = \frac{B L}{R} sin \frac{\phi}{2}

The following equation is the arithmetic average of L(sinβ + sin(α − β)) − Bsinβ for 0 < β < α:


\frac{1}{\alpha} \int_0^{\alpha} L ( sin \beta + sin (\alpha - \beta )) B\beta = \frac{( cos \alpha -1 ) ( B - 2 L )}{\alpha}


\frac{cos \alpha -1}{\alpha} = \frac{B L}{R (B - 2 L)} sin \frac{\phi}{2}

This equation relates α and φ for an average projection.

Fig. 14 The measured angles φ over the torsion angles α .
Fig. 14 The measured angles φ over the torsion angles α .

This graph (fig. 14) shows the result plotted for radius R=6 nm, base length B=35 nm and total length L=98 nm. For this calculation φ was calculated for α from 0 to 133° because of the maximum of ε at this angle. It can be shown in TEM images that the maximum torsion is below this maximum angle α by observing the change of the base width. The maximum change refers to a torsion of 180° and can easily be recognized by thinner base. However, none of the structures examined show a big change of base width.


Observing 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 (fig. 15) 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.

Fig. 15a Side view of BM2 without DNA-binders.
Fig. 15a Side view of BM2 without DNA-binders.
Fig. 15b Side view of BM2 with one EtBr molecule every 7bp.
Fig. 15b Side view of BM2 with one EtBr molecule every 7bp.


Total Twist

The calculations of the base twist result in a modified D for the arm twist. Focusing on the base twist theory the distance between two points on the center lines of the arm cylinders can be described by:


D(z) = 2 \cdot R_B \cdot \left( 1 + \frac{z}{B} \right) \cdot \sqrt{2-2 \cdot cos (\alpha_B)}

Now the arm twist theory can be applied using D(z) instead of the fix value D:


d = 2 \cdot \sqrt{{R_A}^2 + \left( \frac{D(z)}{2} \right)^2 - R_A \cdot D(z) \cdot cos \left( \alpha_A (z) \right)}

The angles for base and arm twist can be normalized to α as an angle per length: \alpha = \frac{\alpha_A}{L-B} = \frac{\alpha_B}{B}. RB is half the distance between the center lines of the arms at the base and RA is the distance of the cylinder center to a point on the cylinder mantle in x-y-direction.


D(z) = 2 \cdot R_B \cdot \left( 1 + \frac{z}{B} \right) \cdot \sqrt{2-2 \cdot cos (\alpha \cdot B)}


d = 2 \cdot \sqrt{{R_A}^2 + \left( \frac{D(z)}{2} \right)^2 - R_A \cdot D(z) \cdot cos \left( z \cdot \alpha + \alpha_0 \right)}

Fig. 16a: Distance of FRET-pair C1→C2 for a counterclockwise (in the view from the arms in direction to the base) torsion.
Fig. 16a: Distance of FRET-pair C1→C2 for a counterclockwise (in the view from the arms in direction to the base) torsion.
Fig. 16b: Distance of FRET-pair C1→C2 for a clockwise torsion.
Fig. 16b: Distance of FRET-pair C1→C2 for a clockwise torsion.


Fig. 16a & 16b correspond to FRET-pair positions C1→C2 as described in the article about FRET-pair positions and to the structure BM14_5/20.

The parameters describing the FRET-pair position are:

  • RA = 3.5nm
  • RB = 6nm
  • α0 = ±17° (+ for counterclockwise, - for clockwise)
  • z = 55nm
  • B = 35nm.

For a Förster-radius R0 = 6.5nm a maximum distance 
d = \sqrt[3]{3} \cdot R_0 \approx 9.4nm
can be evaluated. This corresponds to a maximum angle of 1° per nm or a maximum torsion of 35° over the length of the base and 55° over the rest of the arms for counterclockwise torsion and 1.4° per nm for clockwise torsion.


Thermal Fluctuation of the Arms

Introduction

In order to estimate the width of the broad spread of the data points around the mean angle from the TEM angle measurements, we calculated the width of a Boltzmann distributed fluctuation of the arms. Hereby we assumed the DNA helices to be cylinders with a certain stretch modulus E, persistence length p and to fluctuate independently. Within these measurements we always quantified the angle between the end points of the arms and the other end (at the base).

Derivation

Fluctuation of one Arm

The probability of finding a rod with a 10 helix bundle cross section at a certain angle φ is given by a Boltzmann distribution

 p(\phi - \phi_0) = \frac{1}{Z} e^{- \frac{E(\phi - \phi_0)}{k_B T}}

where the energy is given by

 E(r ) = \frac{1}{2} \frac{E_{DNA} I_{10 hb} L}{r^2}

with r the radius of curvature, L the contour length of the arms and I10 hb the area moment of inertia of a 10 helix bundle. If we assume a constant curvature of the arms, the radius of curvature can be translated in an angle φ (see Fig. 18).

Fig. 18 Fluctuation angle φ.
Fig. 18 Fluctuation angle φ.

 2 \phi = 2π \frac{L}{2\pi r}

 E(\phi- \phi_0) = \frac{1}{2} \frac{4 E_{DNA} I_{10 hb} (\phi - \phi_0)^2}{ L }

The area moment of inertia of a rod is given by

 I = \frac{R^4}{4} \pi

Now we're able to calculate the area moment of inertia of the 10 helix bundle with the parallel axis theorem (see Fig. 19).

 I = \sum_n (I_n + {z_n}^2 A_n) = I_{DNA} (10 + 320 sin (\frac{\pi}{6}))

Fig. 19: Cross section of one of the 10 helix bundle arms. The distances for the parallel axis theorem and the direction of the fluctuation we are looking at are indicated in red.
Fig. 19: Cross section of one of the 10 helix bundle arms. The distances for the parallel axis theorem and the direction of the fluctuation we are looking at are indicated in red.

and therefore the energy needed to bend one arm by the angle φ as well as the probability.

Fluctuation of the Measured Angles

Since the fluctuation of the arms is stochastically independent, we have to take the convolution of the two probabilities:

 p(\Delta\phi) = \int p_1 (\phi_1 ) p_2 (\Delta\phi - \phi_1) d\phi_1

with Δφ = φ2 − φ1

With all this follows

 p(\Delta\phi) = \frac{1}{Z^2}  \sqrt{\frac{π k_B T}{E_{eff}}} e^{-\frac{E_{eff}}{2 k_B T}(\Delta \phi - \phi_0)^2}

with

 E_{eff} = \frac{2 E_{DNA} I_{DNA} (10 + 320 sin(\frac{\pi}{6}))}{L}

Therefore we get a standard deviation of the measured angles of

 \sigma = \sqrt{\frac{k_B T}{E_{eff}}} = \sqrt{\frac{L}{2 p (10 + 320 sin(\frac{\pi}{6}))}}

Double stranded DNA has a persistence length of about p \approx 50 nm and the arms of the U have a length of about  L \approx 98 nm (predicted by the json file). This gives us a width of the distribution of about

 \sigma \approx 4.4^{\circ}

Experimental Considerations for FRET

Basic FRET-Theory

Fig. 20 FRET efficiency over the torsion α
Fig. 20 FRET efficiency over the torsion α

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 EFRETand is a sharply decreasing function of the distance between the dyes. The distance d (as derived from the arm twist theory above) where EFRET is exactly 0.5 is defined as the Förster radius rF. The following equation describes EFRET as a function of the torsion angle of twisted arms according to the arm-twist theory:



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}

For d=rF=6,5 nm (see below), D=13 nm and R=4 nm (both parameters derived from theU'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 6.5 nm 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 setup 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 real time PCR, but can also perform single molecule experiments at our TIRF microscope.

FRET-pair Positions

Fig. 21: FRET-pair positions marked on the cross section.
Fig. 21: FRET-pair positions marked on the cross section.

Since theU 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 theU 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 theU's construction allows for FRET pairs positioned every 21 basepairs, which is visualized in figure 22 below. Symmetric solutions align without X-axis shift, whereas asymmetric solutions would expose a 7 basepair shift.

Fig. 22 Flourophore positions shown in the caDNAno file.
Fig. 22 Flourophore positions shown in the caDNAno file.

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.