Synchronous coefficient of drag alteration

For explanatory purposes consider an electrophoretic particle moving (driven) in an electric field. Let:

${\displaystyle E={\widehat {E}}E_{0}cos(\omega t)}$ (1)

and${\displaystyle dv_{x_{cos2\omega }}}$

${\displaystyle {\overrightarrow {v}}(t)=\mu cos(\omega t){\widehat {E}}E_{0}}$ (2)

denote an electric field and the velocity of the particle in such a field. If ${\displaystyle \mu }$ is constant the time average of ${\displaystyle {\overrightarrow {v}}(t)=0}$.

If ${\displaystyle \mu }$ is not constant as a function of time and if ${\displaystyle \mu }$ has a frequency component proportional to ${\displaystyle cos(\omega t)}$ the time average of ${\displaystyle {\overrightarrow {v}}(t)}$ need not be zero.

Consider the following example:

${\displaystyle \mu (t)=\mu _{0}+\mu _{1}cos(\omega t)}$ (3)

Substituting (3) in (2) and computing the time average, ${\displaystyle {\bar {\overrightarrow {v}}}}$, we obtain:

${\displaystyle {\bar {\overrightarrow {v}}}={\frac {1}{2}}\mu _{1}{\widehat {E}}E_{0}}$ (4)

Thus, it is possible to have the particle experience a non-zero time average velocity, in other words, a net electrophoretic drift, even when the time average of the applied electric field is zero.

Consider a particle under a force field that has a velocity parallel to the field direction and a speed proportional to the square of the magnitude of the electric field (any other non-linearity can be employed[1]):

${\displaystyle {\overrightarrow {v}}=k{\widehat {E}}(E)^{2}}$ (5)

The effective mobility of the particle (the relationship between small changes in drift velocity ${\displaystyle d{\overrightarrow {v}}}$ with respect to small changes in electric field ${\displaystyle d{\overrightarrow {E}}}$) can be expressed in Cartesian coordinates as:

${\displaystyle dv_{x}={\partial v_{x} \over \partial E_{x}}dE_{x}+{\partial v_{x} \over \partial E_{y}}dE_{y}}$ (6)

${\displaystyle dv_{y}={\partial v_{y} \over \partial E_{x}}dE_{x}+{\partial v_{y} \over \partial E_{y}}dE_{y}}$ (7)

Combining (5), (6) and (7) we get:

${\displaystyle dv_{x}=k{\Biggl [}{\Biggl (}E+{\frac {E_{x}^{2}}{E}}{\Biggr )}dE_{x}+{\Biggl (}{\frac {E_{x}E_{y}}{E}}{\Biggr )}dE_{y}{\Biggr ]}}$ (8)

${\displaystyle dv_{y}=k{\Biggl [}{\Biggl (}{\frac {E_{x}E_{y}}{E}}{\Biggr )}dE_{x}+{\Biggl (}E+{\frac {E_{y}^{2}}{E}}{\Biggr )}dE_{y}{\Biggr ]}}$ (9)

Further consider the field E is applied in a plane and it rotates counter-clockwise at angular frequency ${\displaystyle \omega }$, such that the field components are:

${\displaystyle E_{x}=Ecos(\omega t)}$ (10)

${\displaystyle E_{y}=Esin(\omega t)}$ (11)

Substituting (10) and (11) in (8) and (9) and simplifying using trigonometric identities results in a sum of constant terms, sine and cosine, at angular frequency ${\displaystyle 2\omega }$. The next calculations will be performed such that only the cosine terms at angular frequency ${\displaystyle 2\omega }$ will yield non-zero net drift velocity - therefore we need only evaluate these terms, which will be abbreviated ${\displaystyle dv_{x_{cos2\omega }}}$ and ${\displaystyle dv_{y_{cos2\omega }}}$. The following is obtained:

${\displaystyle dv_{x_{cos2\omega }}={\frac {kE}{2}}[cos(2\omega t)]dE_{x}}$ (12)

${\displaystyle dv_{y_{cos2\omega }}={\frac {kE}{2}}[cos(2\omega t)]dE_{y}}$ (13)

Let ${\displaystyle dE_{x}}$ and ${\displaystyle dE_{y}}$ take the form of a small quadrupole field of intensity ${\displaystyle dE_{q}}$ that varies in a sinusoidal manner proportional to ${\displaystyle cos(2\omega t)}$ such that:

${\displaystyle dE_{x}=-dE_{q}xcos(\omega t)}$ (14)

${\displaystyle dE_{y}=dE_{q}ycos(\omega t)}$ (15)

Substituting (14) and (15) into (12) and (13) and taking the time average we obtain:

${\displaystyle {\bar {dv_{x}}}={\bar {dv_{x_{cos2\omega }}}}=-{\frac {kEdE_{q}}{4}}x}$ (16)

${\displaystyle {\bar {dv_{y}}}={\bar {dv_{y_{cos2\omega }}}}=-{\frac {kEdE_{q}}{4}}y}$ (17)

which can be summarized in vector notation to:

${\displaystyle {\bar {d{\overrightarrow {v}}}}=-{\frac {kEdE_{q}}{4}}{\overrightarrow {r}}}$ (18)

Equation (18) shows that for all positions ${\displaystyle {\overrightarrow {r}}}$ the time averaged velocity is in the direction toward the origin (concentrating the particles towards the origin), with speed proportional to the mobility coefficient k, the strength of the rotating field E and the strength of the perturbing quadrupole field ${\displaystyle dE_{q}}$.

To perform SCODA concentration of DNA molecules, the sample must be embedded in the separation media (gel) in locations where the electrophoretic field is of optimal intensity. This initial translocation of the sample into the optimal concentration position is referred to as "injection". The optimal position is determined by the gel geometry and location of the SCODA driving electrodes. Initially the sample is located in a buffer solution in the sample chamber, adjacent to the concentration gel. Injection is achieved by the application of a controlled DC electrophoretic field across the sample chamber which results in all charged particles being transferred into the concentration gel. To obtain a good stacking of the sample (i.e. tight DNA band) multiple methods can be employed. One example is to exploit the conductivity ratio between the sample chamber buffer and the concentration gel buffer. If the sample chamber buffer has a low conductivity and the concentration gel buffer has a high conductivity this results in a sharp drop off in electric field at the gel-buffer interface which promotes stacking.

Once the DNA is positioned optimally in the concentration gel the SCODA rotating fields are applied. The frequency of the fields can be tuned such that only specific DNA lengths are concentrated. To prevent boiling during the concentration stage due to Joule heating the separation medium may be actively cooled. It is also possible to reverse the phase of SCODA fields, so that molecules are de-focused.

As only particles that exhibit non-linear velocity experience the SCODA concentrating force, small charged particles that respond linearly to electrophoretic fields are not concentrated. These particles instead of spiraling towards the center of the SCODA gel orbit at a constant radius. If a weak DC field is superimposed on the SCODA rotating fields these particles will be "washed" off from the SCODA gel resulting in highly pure DNA remaining in the gel center.

The SCODA DNA force results in the DNA sample concentrating in the center of the SCODA gel. To extract the DNA an extraction well can be pre-formed in the gel and filled with buffer. As the DNA does not experience non-linear mobility in buffer it accumulates in the extraction well. At the end of the concentration and purification stage the sample can then be pipetted out from this well.

The electrophoretic SCODA force is gentle enough to maintain the integrity of high molecular weight DNA as it is concentrated towards the center of the SCODA gel. Depending on the length of the DNA in the sample different protocols can be used to concentrate DNA over 1 Mb in length.

DNA concentration and purification has been achieved directly from tar sands samples resuspended in buffer using the SCODA technique. DNA sequencing was subsequently performed and tentatively over 200 distinct bacterial genomes have been identified.[2][4] SCODA has also been used for purification of DNA from many other environmental sources.[5][6]

Mutant DNA (green) is being separated from wild type DNA (red) during sequence specific SCODA.

The non-linear mobility of DNA in gel can be further controlled by embedding in the SCODA gel DNA oligonucleotides complementary to DNA fragments in the sample.[7][8] This then results in highly specific non-linear velocities for the sample DNA that matches the gel-embedded DNA. This artificial specific non-linearity is then used to selectively concentrate only sequences of interest while rejecting all other DNA sequences in the sample. Over 1,000,000-fold enrichment of single nucleotide variants over wild-type have been demonstrated.

An application of this technique is the detection of rare tumour-derived DNA (ctDNA) from blood samples.[9]