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\bigskip

\centerline{\bf
Genotyping by quantitative heteroduplex analysis: 
}
\centerline{\bf
Theoretical derivation and experimental verification of optimal DNA mixing.
}

\bigskip

\centerline{\bf
RA Palais, MA Liew, and CT Wittwer
}

\bigskip

{\bf
\bigskip

Detection of heteroduplexes effectively screens for heterozygotes, but 
usually does not distinguish between different homozygotes.  
If DNA of known genotype is added to PCR products of
a sample of unknown homozygous genotype, and the mixed DNA is 
dissociated then hybridized, formation of heteroduplexes 
indicates that different genotypes are present.
Instead of mixing after PCR, which requires two analyses and
increases the risk of product contamination, we propose adding
DNA of known homozygous genotype to each unknown before PCR.  
If the mixture fraction is chosen carefully, 
the amount of heteroduplexes produced will distinguish
among a heterozygous SNP, a homozygous SNP, and wild-type DNA.
Our analysis suggests that when homozygotes
are most similar, the quantity of additional DNA which 
optimizes the separation of the three genotypes 
of  bi-allelic, diploid DNA comprises one-seventh of
the resulting mixture.
Experimental verification with both high-resolution melting analysis 
and quantitative temperature gradient capillary electrophoresis (qTGCE)
confirmed this prediction,
and the importance of the correct proportion.
If the proportion is one-third or one-half, for instance, some
genotypes are virtually indistinguishable.
When combined with high-resolution melting analysis, this technique
requires only one close-tube analysis 
for full genotyping. 

}

\bigskip

Heteroduplex analysis is a popular technique to screen for sequence 
variants in diploid DNA.  After PCR, heteroduplexes are usually 
separated by conventional gel electrophoresis $^{(1,2,3)}$, although denaturing high 
pressure liquid chromatography (DHPLC), $^{(4)}$ and temperature gradient 
capillary electrophoresis (TGCE) $^{(5)}$ can be used.  Recently, 
heteroduplexes have been detected in solution without separation by 
high-resolution melting analysis.  Either labeled primers $^{(5)}$ or 
a saturating DNA dye $^{(7)}$ were used to detect a change in shape of 
the fluorescent melting curve when heteroduplexes were produced.  
High-resolution melting of PCR products from diploid DNA has been 
used for mutation scanning $^{(8-10)}$, HLA matching $^{(11)}$, and 
genotyping $^{(7, 12)}$.   

\bigskip

Heteroduplex analysis is seldom used for genotyping because different 
homozygotes are usually not separated.  In some cases, DHPLC 
may separate PCR products by size $^{(13)}$.  However, both DHPLC and 
TGCE usually fail to detect homozygous single base changes, 
small insertions and deletions.  If suspected, these homozygous 
changes can be detected by mixing the PCR product with a known 
homozygous PCR product.  However, two sequential analyses are 
required and the concentrated PCR product is exposed to the 
laboratory, increasing the chance of PCR product contamination of 
subsequent reactions. 

\bigskip

In contrast to DHPLC and TGCE, different homozygotes can usually 
be distinguished by high-resolution melting analysis.  
Complete genotyping of human SNPs is possible in over 90\% of 
cases because different homozygotes differ in melting 
temperature $^{(14)}$.  However, in some cases the two homozygotes 
cannot be distinguished and mixing studies are necessary.  
When homozygous samples are mixed after PCR, equal volumes of PCR 
products are combined, denatured, annealed, and melted.  

\bigskip

Alternatively, unknown DNA can be mixed with known homozygous 
DNA before PCR.  If the mixed samples have the same genotype, 
no heteroduplexes will be produced. 
If the mixed samples are not the same, different amounts of heteroduplexes
will be produced, depending on the genotype and
the amount of homozygous DNA added.

\bigskip

Previously, we empirically determined that the optimum amount 
of known homozygous DNA to distinguish all SNP genotypes 
was approximately 15\%.  We now present a rigorous derivation of 
this optimum, by analyzing the theoretical heteroduplex content
and its contribution to melting curves and TGCE measurements
across a full spectrum of genotype mixing proportions, which
are also of interest in pooled sample studies. We then
verify the close agreement of the theory with experiment 
using both high-resolution melting analysis and TGCE.             
The experiments emphasize the importance of using
the correct proportion of added DNA: If the proportion is one-third or one-half rather
than one-seventh, for instance, some genotypes can be virtually 
indistinguishable.

\bigskip


In Fig. 1, we show high-resolution melting curves 
of amplicons from DNA exhibiting three SNP genotypes.
The melting curve corresponding to samples with
a homozygous mutation is indistinguishable from that 
of the wild type, due to nearest-neighbor thermodynamic
symmetry. (This says that the bases
immediately surrounding the mutation are identical
when the strands are interchanged, e.g., 

\smallskip
\noindent
5'-TCA-3' \quad  5'-TGA-3' \hfil\break
3'-AGT-5' \quad  3'-ACT-5' \hfil\break

\smallskip

The melting curves corresponding to 
heterozygous samples, which as genomic
DNA consist of equal parts wild-type homoduplexes
and mutant homoduplexes, appear quite different
than the melting curves of either of these
species of duplex. Even though PCR amplifies
all strands in the form of homoduplexes,
by the time it plateaus, strands are
reassociating randomly into homoduplexes
and heteroduplexes instead of extending.
In the heterozygous case, the DNA that is melted 
is an equal mixture of four species
of duplex, the wild-type and mutant heteroduplexes,
and two types of nearly complementary 
heteroduplexes, so that the total heteroduplex
content is ${1 \over 2}$ (Fig. 2).

\bigskip
There are only complementary strands amplified
from wild-type and homozygous mutant samples,
so even at the end of PCR, there are no
heteroduplexes present.
After PCR and analysis has determined a sample
to be indistinguishable wild-type or mutant 
homozygous, a procedure sometimes known as 
spiking can be performed,
which consists of of adding DNA of known genotype
to determine genotypic identity or difference 
from the absence or presence of heteroduplexes.

\bigskip
Our goal is to find a method 
requiring no post-PCR mixing (which is suceptible
to contamination) which can distinguish wild-type
and homozygous mutant from each other, as well as 
from heterozygous samples, in one-step.
To do so, we seek the optimal fraction of wild-type
DNA to be added to samples before PCR, which we
call the `mixture fraction', so that after 
amplification and random reassociation of strands, 
the heteroduplex content of mixtures with 
the different genotypes will make the resulting
melting curves most distinguishable. 

\bigskip

The mixture with a wild-type samples will still have no
heteroduplex content regardless of the amount of
the identical DNA which is added. 
In contrast, 
if wild-type DNA is mixed with a homozygous mutant sample,
even though PCR amplifies all strands of the mixture
as homoduplexes, by the time it plateaus (or after
heating then cooling the mixture to promote
random reassociation) a fraction of
heteroduplexes will be formed,
depending on the amount of wild-type DNA added.
If wild-type DNA is mixed with
a heterozygous mutant sample, the mixture will
now consist of 
unequal parts of wild-type homoduplexes and mutant homoduplexes.
At the end of PCR, or after dissociating 
by heating and annealing by cooling, a reduced fraction of
heteroduplexes will be formed depending on the amount of 
wild-type DNA added.
As the heteroduplex enhanced 
homozygous mutant melting curve moves away from
the wild-type melting curve, the heteroduplex reduced
heterozygous melting curve moves toward them both. We 
seek the point where the three are best separated.


\centerline{\bf MATERIALS AND METHODS }

\bigskip

Our basic method is to derive a model for the heteroduplex
content of a mixture of genotypes after amplification
and reassociation, in terms of the mixture fraction.
We then develop quantitative models for the melting curve
of such a mixture in terms of the melting
curves of the constituent duplex types and the heteroduplex
content, and similarly for TGCE arrival data of mixtures
in terms of their constituent duplex types and content.

\bigskip

In the case described above when the homozygous mutant and 
wild-type curves are indistinguishable,
the result of our analysis is that the maximum separation between 
melting curves of different genotypes is given by difference in
the heteroduplex content of the mixtures times
the maximum separation between the common homoduplex melting curve
and the average heteroduplex melting curve.

\bigskip
After presenting the model,
we use it to optimize the separation of heteroduplex contents 
of the three genotypes in terms of mixture fraction.
Next we present our materials for performing experimental tests 
of the model and methods for their analysis before
discussing the results in the final section.

\bigskip

We begin in Table 1a which lists the type and fraction
of homoduplexes are amplified when we mix wild-type and 
unknown DNA in fraction $x$ of wild-type to $1-x$ 
of either 
wild-type, homozygous mutant, or heterozygous mutant.
We refer to $x$, which measures the fraction of 
wild-type DNA added as a proportion of the mixture of 
sample plus added DNA, as the `mixture fraction'.
When $x=0$, the mixture consists of just the original
genotype. When $x=1$, it describes
a pure wild-type sample regardless of the initial
unknown genotype. 
Table 1b lists the type and fraction of duplexes
which result when complementary
and nearly complementary strands reassociate
independently to form duplexes 
when PCR has plateaued, or after subsequent heating
and cooling.
The assumption of independence states that
complementary and nearly complementary
strands associate in proportion to their
concentration, so we obtain the fractions in
Table 1b simply by multiplying the corresponding
fractions from Table 1a.

\bigskip
Corresponding to each duplex species $d_j$ will be its
normalized
fluorescence curve,
$F_j(T)$.
The function $F_j(T)$ refers to the curve with  background
fluorescence removed and accounts
for factors such as lower fluorescence
per duplex behavior of heteroduplexes.
Table 1c gives
the normalized melting curves for the mixture fraction
$x$ of wild-type DNA and $1-x$ DNA sample of each of the genotypes,
Assuming fluorescence is additive,
We use $W(T,x)$ for wild-type sample,
$M(T,x)$ for mutant homozygous sample, and $H(T,x)$ for
heterozygous sample ($T$ is temperature).
Experimental examples of the unmixed curves $W(T,0),M(T,0),$ and $H(T,0)$ 
are shown in Fig. 1.  and theoretical examples of 
$F_1=W(T,0), F_2, F_3, F_4=M(T,0)$, and $H(T,0)$
are shown in Fig. 2.

\bigskip

Our goal is to find the mixture fraction $x$ which maximizes our ability 
to separate the three mixed melting curves. We use
the maximum vertical distance between
two graphs to measure their separation, and the minimum
separation among any of the three pairs of curves to score
the effectiveness of our mixture fraction. Mathematically then,
we seek


$$\max_{x \in [0,1]} 
\min \{ 
\max_T | W(T,x) - M(T,x) | ,
\max_T | W(T,x) - H(T,x) | ,
\max_T | H(T,x) - M(T,x) | 
\}.\eqno(1)$$

Table 1d gives expressions for the
differences $W-M$ and $W-H$, and after simplifying,
we will determine $H - M$ as the difference of these 
differences.

\bigskip

Since we are most interested in the situation where 
$F_1(T)$ and $F_4(T)$ are difficult to distinguish,
or we would not be adding DNA of one type to do so,
we assume $F_1(T)=F_4(T)$.
This condition is predicted by the nearest-neighbor 
approximation for SNPs with the symmetry described 
in the previous section.
Although the two homoduplexes have the same melting curves
in the symmetric case, the heteroduplex species typically 
have very different thermodynamic behavior, 
and fortunately our derivation does not require
that $F_2$ and $F_3$ agree.
Table 1e gives simplified forms for the
difference curves in this case,
indicated by the subscript $\ _=$.
They are obtained by first
replacing  $F_4$ by $F_1$, combining terms 
removing a common factor from the resulting
coefficients of $F_1, F_2$, and $F_3$, then
writing $F_1={F_1 + F_4 \over 2}$.

\bigskip

The result justifies our earlier statement that
when the homozygous mutant and 
wild-type curves are indistinguishable,
the difference between melting curves of different
genotypes is given by difference in
the heteroduplex content of the mixtures times
the difference between the common homoduplex melting curve
and the average heteroduplex melting curve:

$$W_=(T,x)- M_=(T,x) = m(x)F(T) \eqno(2a)$$

$$W_=(T,x)- H_=(T,x) = h(x)F(T) \eqno(2b)$$

and therefore

$$H_=(T,x)- M_=(T,x) = (h(x)-m(x))F(T), \eqno(2c)$$

where

$$F(T) =  {F_1(T)+ F_4(T) \over 2} -  { F_2(T) + F_3(T) \over 2} \eqno(3)$$

is the difference between the average homoduplex melting curve
and the average heteroduplex melting curve, and

$$m(x)=2x(1-x) \eqno(4a)$$
$$h(x)= { 1-x^2 \over 2} \eqno(4b)$$
$$h(x)-m(x)= {3x^2-4x+1 \over 2}. \eqno(4c)$$

represent the total heteroduplex content difference
between the various genotype mixtures with wild-type DNA.
The graphs of the absolute values of the three heteroduplex
content difference functions are given in
Fig. 3, annotated with key features derived below.
The (theoretical) standard difference curve $F(T)$ is
shown in Fig. 2.

\bigskip
The expressions for $m(x),h(x),$ and $h(x)-m(x)$
uncouple the $x$ (mixture fraction of added DNA) and $T$
(temperature) dependence of the differences among fluorescence
curves of different genotypes, and identify
the $x$ dependence of the separation of melting curves
of different genotypes as proportional solely to 
the difference in heteroduplex content of the mixtures.
This allows us to optimize
the mixture fraction independently, regardless of the
specific nature of individual duplex curves contributing
to the superpositions. 
This is perhaps surprising, since in general, different 
combinations of different functions 
result in varying shapes and locations of their extrema.
Since all expressions are
linear, we can obtain the difference of the
negative derivative curves of the different genotypes
simply by replacing $F_j$ in the expressions
above with their negative derivatives, and all of the results we will obtain
regarding maxima and minima will still hold.

\bigskip

Our original genotype separation problem has become 
$$ \max_{x \in [0,1]} \min \{ m(x)  , h(x)  , | h(x) - m(x) | \}  F ,\eqno(5)$$
where
$$F= \max_{T} F(T) \eqno(6)$$
represents the twice the separation of the original (no added wild-type
DNA) heterozygous
curve from the common original wild-type and homozygous SNP curves,
since $h(0)={1 \over 2}$.
So all that remains to optimize the mixture fraction $x$ is
to maximize the separation by maximizing the
smallest of the three heteroduplex
content differences:
$$s(x)= \min \{ m(x)  , h(x)  , | h(x) - m(x) | \} . \eqno(7)$$ 
In Fig. 3, this corresponds to finding 
the highest point on the lowest of the 
three graphs of $m$, $h$, and $|m-h|$.
Table 1f identifies three intervals on which
the ordering of these functions is preserved.
We can find these intervals by finding the
points where two of them become equal:
$h(x)=m(x)$ when
$h(x)-m(x)={3 x^2 -4x + 1 \over 2}= {1 \over 2}(3x-1)(x-1)=0$,
at $x=1/3$ and $x=1$,
and
$m(x)=h(x)-m(x)$ when
$h(x)-2m(x)=
{7 \over 2}x^2 -4x +{1 \over 2}={1 \over 2}(7x-1)(x-1)
=0$, at $x={1 \over 7}$ and $x=1$.

\bigskip
Calculus theorems confirm our visual intuition that the maximum of
of the lowest graph can only occur where the slope of
its tangent is zero, or it does not have a well-defined
tangent. (If $s(x)$ has a local extremum at $x=a$, then
$s'(a)=0$ or $s'(a)$ does not exist.)
The only place $s'(x)=0$ is
halfway between its roots ${1 \over 3}$ and $1$ (since
it is quadratic in this interval), i.e.,
at $x={2 \over 3}$.
This corresponds to adding twice as much wild-type DNA
as there was unknown DNA and gives a 
separation of ${1 \over 6}F$, 
between the heterogygous and homozygous SNP curves,
or ${1 \over 3}$ of the original separation between 
the heterozygous melting curve and the other two.
The separation between the wild-type 
melting curve and the other two melting curves
will be larger.

\bigskip

The only places $s(x)$
is not differentiable is where it
changes form, i.e., at $x={1 \over 7}$ and $x={1 \over 3}$.
Comparing the values at these points and
$x={2 \over 3}$, we find the
optimal mixture fraction occurs at 
$x=x_*={1 \over 7}$, as indicated in Fig. 3.
Here, the melting curve of the mixture with homozygous DNA 
is halfway between the other curves at the temperature 
of maximum separation, where 
its separation from each is ${12 \over 49} F$.
This is only barely less than half of the 
original separation of ${1 \over 2}F = {24 \over 48}F$
between the heterozygous melting curve and the other two 
genotype curves.

\bigskip

A simple heuristic explanation for this value,
corresponding to adding one part wild-type DNA to 
six parts unknown sample
is based upon the observation that the melting
curves will be optimally separated when the homozygous
mutant curve becomes equidistant from both the wild-type
and heterozygous melting curves, so the heteroduplex
content of the wild-type-heterozygous mixture
must be exactly twice that of the 
wild-type-homozygous mixture: $h(x)=2m(x)$ 
The ratio of $1$ part wild-type to $6$ parts unknown is
optimal because $6$ is the unique number which can be
divided in equal parts ($3+3$, the heterozygous sample
strands), and when one of the parts ($3$, SNP strands) 
is multiplied by the other plus $1$ ($4=3+1$, wild-type sample 
plus added wild-type strands) one obtains exactly twice the 
original number ($6$, the homozygous SNP strands) multiplied 
by $1$ (added wild-type strands.)
At the simplest level, it is because  
$({6 \over 2})({6 \over 2}+1)=2(6)(1)$ that $x_*={1 \over 7}$
of the total unknown plus added DNA is the optimal 
mixture fraction.  This is visualized in the animation

\bigskip

http://www.math.utah.edu/$\sim$palais/pcr/michael/spike/spike.html

\bigskip


When the wild-type and homozygous SNP curves are not the
same, the max-min problem does not separate into
individual mixture fraction and temperature problems.
In this case, the optimal mixture fraction still
may be characterized by the two-dimensional generalization
of the above criteria, but it must be found numerically
depending upon the specific curves $F_j(T)$. That is,
the mixture fraction-temperature rectangle $[0,1] \times [T_1,T_2]$
must be divided into regions on which the corresponding
$h(x,T)-m(x,t)$ is either positive or negative, and
within these regions, local extrema are characterized
by when the gradient of the
smaller of $h(x,T)$ and $m(x,T)$, denoted
$s(x,T)$ or of the larger minus the smaller, denoted $l(x,T)-s(x,T)$
is zero, or when $s(x,T)=l(x,T)-s(x,T)$. 
It seems reasonable to expect that the best separation
could be obtained by including the possibility
of adding whichever type of homozygous SNP or
wild-type curve is already closer to the heterozygous curve.

\bigskip

To test the model of the previous section, we performed several experiments 
to genotype DNA for the presence of a homozygous or heterozygous 
SNP, 187C$>$G, which is found in the hemochromatosis gene (HFE). This 
mutation we analyzed 

\smallskip
\noindent
5'-TCA-3' \quad  5'-TGA-3' \hfil\break
3'-AGT-5' \quad  3'-ACT-5' \hfil\break

\smallskip

has the nearest-neighbor symmetry described above which results in a 
melting curve for the homozygous case which is theoretically identical 
to and experimentally indistinguishable from that of wild-type DNA.

\bigskip

(Table 2 contains the complete sequence of melting analysis and TGCE
amplicons with SNP and primers highlighted.
H63D\_sequence\_031010.doc)


\bigskip

We examined mixtures of known quantities of DNA of each genotype 
and additional wild-type DNA, 
in 21 ratios of the added DNA to
total DNA (initial plus added.) The ratios, 
from 1/28 to 14/28 by increments of 1/28, and from
15/28 to 27/28 in steps of 2/28, allowed us to include the 
theoretically optimal value, 1/7, and observe the behavior of the 
process in some detail over a wide
range of interest for pooled samples as well. The ratio j/28 of 
additional wild-type DNA to total DNA (unknown sample plus additional)
corresponds to the ratio j/(28-j) of additional DNA to unknown, 
so for instance,
the optimal value of 4/28 of total DNA
corresponds to  adding 4/24=1/6 as much wild-type as there is
unknown sample DNA.


\bigskip

We mixed wild-type DNA in three replicates for each of the three genotypes 
(denoted WT, MUT, and HET) before PCR with the appropriate
mixture fraction of wild-type DNA.
All samples with a common mixture fraction were amplified together
in the presence of a high-resolution fluorescent dye,
along with two control samples containing heterozygous DNA
with no wild-type added.
Since the extension step of each PCR cycle generates only
homoduplexes, after amplification is complete, we perform
a final additional melt and reannealing to produce the heteroduplexes.
Samples with the same mixture fraction of additional DNA were also
analyzed simultaneously.

\bigskip

Detailed protocols for the amplification and subsequent heating
and cooling before melting analysis are given in Table 3.

\bigskip

We performed high-resolution melting analysis on all of the 
resulting samples to produce actual fluorescence vs. temperature 
melting curves corresponding to the model functions of the 
previous section. 

\bigskip

This is a closed-tube process which avoids risk of contamination and 
leaves the sample undisturbed for further types of analysis.
It provides a fast, economical, and accurate method of genotyping and 
mutation scanning which has been described and studied in a variety of 
contexts $^(7-12)$.

\bigskip

Melting curves are first standardized by removal of background
fluorescence. Next, they are temperature shifted to adjust for small
variations in reported temperature, by
superimposing the `toe' feature common to
all curves, where only the most stable homoduplexes are left to melt.
Difference plots were used to
highlight relative variation between genotypes. 

\bigskip

The value and location
of the maximum 
difference and the area 
between curves of the different genotypes and the average of the
wild-type replicates were recorded for analysis and comparison
with the theory.
According to the theory, location of maximum difference is constant, 
and magnitude of maximum difference, and area under difference 
are directly proportional to the heteroduplex concentration of the samples. 

\bigskip

To obtain an independent and more direct analysis of
heteroduplex content,
we also performed temperature-gradient 
capillary electrophoresis (TGCE) on 
a set of genotype mixtures representing the
same range of mixture fractions. 
In this technique, 
the arrival of duplexes in a sample is detected after they are drawn through 
a gel. Each species of duplex has a characteristic arrival time
distribution depending on its spatial conformation.
This corresponds to the frequency of duplex arrivals per frame, the
quantity which is actually measured. The 
center of heteroduplex peaks are significantly delayed 
and separated from each other in comparison to homoduplex
peaks.
This is due to the distinct `bubbles' formed by different mismatched 
base pairs of the two heteroduplexes. The two species 
of homoduplex have no bubbles and appear as one peak.
The two heteroduplex peaks are easily separated from the homoduplex 
peak and from each other. These peaks exhibit simple mathematical 
behavior which makes it possible to separate and quantify the relative 
contributions of the heteroduplexes. 
This permitted us to more directly validate 
the relative concentration of heteroduplexes in samples
upon which our theoretical model 
of mixture separation is based.

\bigskip

For TGCE analysis, we used
larger amplicons containing the same region
in the HFE gene. In addition to larger amplicons providing 
better TGCE results, this also could show that our results 
were independent of amplicon size.
Detailed protocols for the amplification of samples for
TGCE analysis are given in Table 3b, and protocols for the
analysis itself in Table 3c.

\bigskip

The TGCE data was analyzed quantitatively as follows.
Individual TGCE peaks were approximated by 
exponential distributions of the form 
$F(t)=Ae^{-kt}, t \geq  t_0; F(t)=0, t<t_0$. Higher resolution 
data might be amenable to closer fit by higher order gamma 
distributions of which the exponential distribution is a special 
case, but since the peaks are only resolved by on the order of 
10 data points, the simplest version must suffice. Some additional 
evidence that this is reasonable is provided by the fact that the 
fit parameters of each peak remained nearly invariant when the window 
of points
used for the fit was varied in size and distance from the peak. 
The observed arrival frequency before each peak did not have the 
strict cutoff behavior of the exponential distribution, as some 
increase above background was seen one frame before the maximum 
of the first arrival peak. However, no increase above background 
could be seen two frames before the first arrival peak. Based upon 
this model, we could solve for the combined amplitudes and decay 
rate of homoduplex concentrations contributing to the first arrival 
peak, and by successive subtraction, iteratively solve for the 
amplitudes of subsequent peaks. 
Because of the large dynamic range of the peaks and 
their narrow extent in terms of data acquisition frames, the 
quantitative results might be expected to
be sensitive to the fitting process. 
For example, we approximated the start of each exponential 
sub-distribution with the frame of the
maximum measured value, even though actual peak is located
somewhere between this frame and an adjacent one.
In spite of this sensitivity,
we found that the decay rates of different peaks 
were nearly independent of duplex species, peak amplitude,
fitting window and method, which provided additional confidence
in the model.
We are investigating more sophisticated gamma fits of the data, 
and corresponding deconvolution techniques which could reduce 
these sources of error. 

\bigskip
Once the constituent peak amplitudes were quantified,
the heteroduplex proportion was determined by first computing the
ratio of the sum of the derived amplitudes of the two heteroduplex
peaks to the sum of these plus the amplitude of the
combined homoduplex peak. This ratio was
then adjusted by a factor close to $1$
which made the same ratio determined from heteroduplex 
control samples
equal to the expected value of $0.5$.



\bigskip

\centerline{\bf DISCUSSION }

\bigskip

Fig. 4 shows the average calculated values of the
maximum difference between the three replicate
melting curves of mixtures with homozygous and heterozygous genotypes
and the average of the wild-type melting curves,
as a function of the mixture fraction.
The values are normalized by a scaling
which makes the value of pure heterozygous control samples equal to $0.5$.
This corresponds to the concentration of heteroduplexes in the theoretical 
model, which is superimposed on the figures. 
The squared correlation between the experiment and the model,
$R^2 ={ (\X \cdot \M )^2 \over (\X \cdot \X ) (\M \cdot \M ) }$
where $\X$ is the vector of experimental values 
and $\M$ is the vector of model values
at the measured mixture fractions.
(We do not use the usual formula for $R^2$ in terms
of sums of squares,
$1 - { \E \cdot \E   \over  \X \cdot \X } $
where $\X + \E = \M$,
which requires $\M$ to be the best fit of the experimental
data satisfying the orthogonality condition
$||\M||^2 + ||\E||^2 = || \X || ^2$, or
$(\M \cdot \E)=0$.
Even though $\M$ is not the best fit, the squared correlation
coefficient values satisfy
$R^2>.99$ for the heterozygous samples, and $R^2>.98$ 
for the homozygous samples.

\bigskip
Fig. 5a shows the standardized melting curves corresponding
to the optimal $4/28$ mixture fraction. 
The replicates cluster indistinguishably, appearing as one curve, 
and the three genotypes are equally separated.
They may easily be classified by the observer's 
eye or by automatic classification  software.
This is a vast improvement from the Fig. 1 in which 
replicates of the homozygous SNP and the wild-type samples overlapped 
each other completely.

\bigskip

For comparison, Fig. 5b and 5c show the standardized
melting curves corresponding to mixture fractions $9/28$
and $14/28$,
in which it is again difficult to distinguish
the homozygous and heterozygous samples as our model
predicts. Fig. 5d show the melting curves
at mixture fraction $19/28$ near where they are again
best separated, albeit in a different order.
This demonstrates
the importance of the correct mixture fraction. Using equal proportions,
or just any small proportion such as $1/3$ gives no improvement
on the original results with no mixing at all!

\bigskip
Fig. 6 shows experimental difference curves between
wild-type melting curves and the other two genotype melting
curves at two values of $x$, $x=0$, or no added DNA (Fig. 6a)
and at the optimal value $x=4/28$ (Fig. 6b). In both case,
the shape and magnitude agree nicely with the theoretical
melting curves and the  predictions of our model.

\bigskip

Fig. 7a shows typical raw TGCE data for three replicates
of mixtures with each genotype with mixture fraction $9/28$,
over a range of frames beginning with frame $1000$
and containing all of the peaks.
Fig. 7b shows the same data normalized by shifting all peaks 
to the same frame number and scaling to the same height.
This figure demonstrates the common heteroduplex content of the
homozygous mutant and heterozygous genotypes at this
mixture fraction, which is responsible for 
their overlapping melting curves seen in Fig. 5b,
reemphasizing the sensitivity of the results to 
choosing the added DNA proportion suboptimally.


\bigskip

Fig. 8 shows the average of the calculated values of the
heteroduplex proportion
of of three replicate of mixtures with homozygous 
and heterozygous samples as a function of the mixture fraction.
Once again, the squared correlation coefficients between the
experimental data and the model are high, with $R^2> .97$
for both heterozygous and homozygous samples.
This agreement between 
the results of a fairly simple analysis and those of the 
melting curve experiments and the theory suggest that  quantitative 
TGCE (qTGCE) estimation of heteroduplex content of mixed or pooled 
samples is indeed feasible and informative.

\bigskip

The experimental results confirm the main points of 
the theory.
The maximum difference between melting curves 
and the heteroduplex concentrations inferred 
from TGCE experiments agree with each other and with 
the theoretical predictions of heteroduplex concentration 
with considerable accuracy over a wide range of 
mixture fractions. 
The area between melting curves and
the location of the maximum difference between 
curves also behave as predicted.
The plots of these quantities follow the quadratic behavior of the 
model qualitatively over the entire range, and are quantitatively 
close over a range of mixture fractions up to 
one-half (14/28) of the total.

\bigskip

Where the data deviates from the model above this mixture fraction, 
there is a definite trend for heteroduplex concentrations estimated 
from TGCE and corresponding melting curve differences to be larger 
than those predicted by the theory for a given mixture fraction. 
Because the heteroduplex 
concentration vs. mixture fraction curves for both the heterozygous 
and heterozygous unknowns are decreasing for mixture fractions greater 
than 14/28, the inferred experimental values correspond to mixture 
fractions lower than those we prepared experimentally. So one 
possible source of such a trend could be that the actual fraction 
of additional wild-type DNA fell short of the intended value as that value 
grew beyond one-half.  Selective amplification (unequal efficiencies) 
in PCR or amplification of initial variations that diminish final 
concentration of wild-type DNA at higher concentrations could 
have such an effect. 
If complementary and nearly-complementary strands anneal preferentially
rather than independent of their differences, 
the assumptions of the model would be violated,
although it would 
be surprising if such a bias favored more rather than less
heteroduplex formation.


\bigskip

Regardless of these subtle deviations from close agreement 
with a fairly simple model, the ultimate test of our method is 
given by the ease with which the simple melting curve approach to 
can be used to genotype the optimally mixed samples, in contrast 
to unmixed or non-optimally mixed ones.

\vfill \eject


\centerline{\bf
References
}

\bigskip

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\medskip
\bye

spectrumedix TM and reference?
