\magnification = \magstep1
\overfullrule = 0pt
\def\degrees{{$^\circ$}}
\def\E{{\bf E}}
\def\X{{\bf X}}
\def\M{{\bf M}}

\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

\centerline{\bf
Abstract
}

\bigskip

Detection of heteroduplexes effectively screens for heterozygotes, but 
usually does not distinguish between different homozygotes.  
If we add DNA of known genotype to PCR products of
a sample of unknown genotype, and the mixture is melted and reannealed,
the absence or presence of heteroduplexes will indicate whether
what was added had the same or different genotype as the sample. 
Depending upon the proportions mixed, the heterozygote still 
could be distinguished from both homozygotes by quantifying
the amount of heteroduplexes produced.
Instead of mixing post-PCR, increasing risk of product 
contamination and requiring two analyses, we propose adding
DNA of known homozygous genotype to each unknown before 
PCR.  Our analysis suggests that when homozygotes
are most similar, adding DNA of known genotype to DNA of
unknown genotype to comprise one-seventh of the total
optimizes the separation of the three genotypes 
of  bi-allelic, diploid DNA.  
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 
is needed for full genotyping. (200)


\bigskip

\centerline{\bf
Introduction
}

\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), although denaturing high 
pressure liquid chromatography (DHPLC, 2) and temperature gradient 
capillary electrophoresis (TGCE, 3) can be used.  Recently, 
heteroduplexes have been detected in solution without separation by 
high-resolution melting analysis.  Either labeled primers (4) or 
a saturating DNA dye (5) 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 (6-8), HLA matching (9), and 
genotyping (5, 10).   

\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 (11).  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 (10).  However, in some cases the two homozygotes 
cannot be distinguished and mixing studies are necessary.  
When samples are mixed after PCR, equal volumes of PCR 
products are combined, denatured, annealed, and melted.  
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. 

\bigskip


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


\centerline{\bf
Experimental tests of the quantitative model.
}

\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 

\bigskip

\noindent
5'-TCA-3' $\to$ 5'-TGA-3'
\smallskip
\noindent
3'-AGT-5' $\to$ 3'-ACT-5'

\bigskip

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

(See Table 1 for 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 
spike to spike plus
unknown corresponds to the ratio j/(28-j) of spike to unknown, 
so for instance,
the optimal value of 4/28 is the same as 4/24=1/6 spike to unknown.


\bigskip

We spiked three replicates of each of the three genotypes 
(denoted WT, MUT, and HET) before PCR with the appropriate
fraction of additional 
wild-type DNA to achieve the desired spike as a proportion of the total. 
All samples with a common spike fraction were amplified together
in the presence of a high-resolution fluorescent dye,
along with two control samples containing unspiked heterozygous DNA.
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 a common spike proportion were also
analyzed simultaneously.

Detailed protocols for the amplification and subsequent melting
and reannealing before melting analysis are in ?

\bigskip

\centerline{\bf  Validation with melting curve data }

\bigskip

Next, 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 (5-10).

\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

Figure 3 shows the average calculated values of the
maximum difference between the three replicate 
spiked homozygous 
and heterozygous SNP melting curves 
and the average of the spiked wild-type melting curves 
as a function of the proportion 
of the spike in the total mixture. The values are normalized by a scaling
which makes the value of unspiked 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 spiking values.
(We cannot use the common alternate formula for $R^2$ in terms
of sums of squares,
$1 - { \E \cdot \E   \over  \X \cdot \X } $
where $\X + \E = \M$.
This 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
Figure 4a shows the standardized melting curves corresponding
to the optimal $4/28$ spiking value. 
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 initial figure in which 
replicates of the homozygous SNP and the wild-type samples overlapped 
each other completely.

\bigskip

For comparison, Figure 4b and 4c show the standardized
melting curves corresponding to spiking values $9/28$
and $14/28$,
in which it is again difficult to distinguish
the homozygous and heterozygous samples as our model
predicts. Figure 4d show the melting curves
at spiking value $19/28$ near where they are again
best separated, albeit in a different order.
This demonstrates
the importance of correct spiking. Using equal proportions,
or just any small proportion such as $1/3$ gives
no better results than no spiking at all!


\bigskip

\centerline{\bf  Validation with quantitative TGCE data }

\bigskip

We also performed temperature-gradient 
capillary electrophoresis (TGCE) on each sample to provide
an independent and more direct analysis of
heteroduplex content.
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. 
We used this to validate our theoretical model of melting curve 
separation, which was based upon relative
concentration of heteroduplexes in the samples.
Figure 5a shows typical raw TGCE data for three replicates
of each genotype spiked with $9/28$ spike.
over a range of frames beginning with frame $1000$
and containing all of the peaks.
Figure 5b 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
spike proportion, which is responsible for 
their overlapping melting curves seen in figure 4b,
reemphasizing the sensitivity of the results to 
choosing the added DNA proportion suboptimally.

\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

Figure 6 shows the average of the calculated values of the
heteroduplex proportion
of three replicate spiked homozygous 
and heterozygous samples as a function of the proportion 
of the spike in the total mixture. 
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 spiked or pooled 
samples is indeed feasible and informative.

\bigskip



\bigskip

\centerline{\bf
Discussion of the results
}

\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 
spiking proportions. 
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 spike proportions up to 
one-half (14/28) of the total.

\bigskip

Where the data deviates from the model above this spiking value, 
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 spike proportion. 
Because the heteroduplex 
concentration vs. spike proportion curves for both the heterozygous 
and heterozygous unknowns are decreasing for spike proportions greater 
than 14/28, the inferred experimental values correspond to spike 
proportions lower than those we prepared experimentally. So one 
possible source of such a trend could be that the actual proportion 
of wild-type spike 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 spike 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 spiked samples, in contrast 
to unspiked or non-optimally spiked ones.

\vfill \eject


\centerline{\bf Appendix: Experimental protocols }

\bigskip
DNA was extracted using a QIAamp DNA Blood Kit (QIAGEN,  Inc., Valencia, 
CA), concentrated by ethanol precipitation and quantified by A260.  

\bigskip


For high-resolution melting analysis, we used small amplicon melting
with primers as close to the SNP as dimer and misprime constraints
permit, as described in Liew et al. (2004) (12).
The PCR protocol followed here was modified slightly from the 
protocol described in (12). The amplicon was 40bp 
long. PCR was performed in a LightCycler with reagents commonly used in 
clinical laboratories.  Ten microliter reaction mixtures consisted of 25ng 
of genomic DNA, 3 mM MgCl2, 1x LightCycler FastStart DNA Master Hybridization 
Probes master mix, 1x LCGreen Plus, 0.5 $\mu$M forward (CCAGCTGTTCGTGTTCTATGAT ) 
and reverse (CACACGGCGACTCTCAT) primers and 0.01U/$\mu$l Escherichia coli 
(E. coli) uracil N-glycosylase (UNG, Roche).  The PCR was initiated with a 
10 min hold at 50\degrees C for contamination control by UNG and a 10 min 
hold at 95\degrees C for activation of the polymerase.  Rapid thermal 
cycling was performed between 85\degrees C and the annealing temperature 
at a programmed transition rate of 20\degrees C/s for 40 cycles.  Samples 
were then rapidly heated to 94\degrees C and cooled to 40\degrees C 
followed by melting curve analysis between 60\degrees C and 85\degrees C to 
confirm the presence of amplicon. 

\bigskip

Prior to analysis on the HR1, samples 
were again rapidly heated to 94\degrees C and cooled to 40\degrees C to 
promote heteroduplex formation. 

\bigskip


Following amplification, an additional melting was performed to denature 
the perfectly complementary post-extension duplexes after which the 
temperature was rapidly decreased to re-anneal strands independent of 
the presence or absence of a single mismatched base-pair. 


\bigskip


PCR protocol for HFE amplification for Temperature Gradient 
Capillary Electrophoresis (TGCE) analysis

\bigskip

The PCR protocol followed here was modified slightly from the protocol 
described in Bernard et al. (1998) (13). The amplicon was 242bp long. 
PCR was performed in a Perkin Elmer 9700 block cycler with similar 
reagents to those used for amplification in the LightCycler.  Ten 
microliter reaction mixtures consisted of 25ng of genomic DNA, 
3 mM MgCl2, 1x LightCycler FastStart DNA Master Hybridization Probes 
master mix, 0.4 $\mu$M forward (CACATGGTTAAGGCCTGTTG) and reverse 
(GATCCCACCCTTTCAGACTC) primers and 0.01U/$\mu$l Escherichia coli (E. coli) 
uracil N-glycosylase (UNG, Roche). All samples were then overlayed 
with mineral oil to prevent evaporation. The PCR was initiated with a 
10 min hold at 25\degrees C for contamination control by UNG and a 6 
min hold at 95\degrees C for activation of the polymerase. Thermal cycling 
consisted of a 30s hold at 94\degrees C, a 30s hold at 62\degrees C and 
a 1min hold at 72\degrees C for 40 cycles followed by a 7min hold at 
72\degrees C for final elongation.  

\bigskip

Upon completion of these thermal 
cycles the samples were then heated to 95\degrees C for 5min followed 
by a slow cool over approximately 60min to 25\degrees C. 
\bigskip

TGCE analysis

\bigskip

The protocol followed here is similar to that described in 
Margraf et al. (2004) (14). To prepare samples for TGCE analysis, 
PCR amplicons were transferred to 24 well TGCE trays and diluted 1:1 
with 1xFastStart Taq polymerase PCR buffer (Roche). These samples were 
then overlayed with mineral oil and the trays loaded into the TGCE 
instrument. TGCE was performed on a commercial instrument 
(Reveal

(TM-melting temperature or trademark?)

mutation discovery system, reagents and Revelation software by 
SpectruMedix LLC, State College, PA) (6).  DNA samples were injected 
electro-kinetically at 2 kV for 45 seconds, resulting in peak heights 
ranging from 5,000-40,000 intensity units with ethidium bromide 
staining. Optimal results were obtained when the temperature was 
ramped from 60-65\degrees C over 21 minutes and data was acquired over 
35 minutes.  Sequential camera images were converted to plots of 
image frame number (time) versus intensity units (DNA concentration).



\vfill \eject

\centerline{\bf
References
}

\bigskip

\noindent
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toolbox and evaluation of heteroduplex analysis.  
Electrophoresis. 1999 Jun;20(6):1186-94.


\medskip

\noindent
2.  Xiao W, Oefner PJ.  Denaturing high-performance liquid 
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\medskip

\noindent
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\medskip

\noindent
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Amplicon melting analysis with labeled primers: a closed-tube method 
for differentiating homozygotes and heterozygotes.  
Clin Chem. 2003 Mar;49(3):396-406.

\medskip

\noindent
5.  Wittwer CT, Reed GH, Gundry CN, Vandersteen JG, Pryor RJ.  
High-resolution genotyping by amplicon melting analysis using 
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\medskip

\noindent
6.  Reed GH, Wittwer CT.  Sensitivity and specificity of 
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\medskip

\noindent
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\medskip

\noindent
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\medskip

\noindent
9.  Zhou L, Vandersteen J, Wang L, Fuller T, Taylor M, Palais B, Wittwer CT.  
High-resolution DNA melting curve analysis to establish 
HLA genotypic identity.  Tissue Antigens. 2004 Aug;64(2):156-64.

\medskip

\noindent
10.  Liew M, Pryor R, Palais R, Meadows C, Erali M, Lyon E, Wittwer C.  
Genotyping of single-nucleotide polymorphisms by high-resolution 
melting of small amplicons.  Clin Chem. 2004 Jul;50(7):1156-64.

\medskip

\noindent
11.  Kuklin A, Davis AP, Hecker KH, Gjerde DT, Taylor PD.  
A novel technique for rapid automated genotyping of DNA polymorphisms 
in the mouse.  Mol Cell Probes. 1999 Jun;13(3):239-42.

\medskip
\noindent
12.	Liew, M., et al., Genotyping of single-nucleotide polymorphisms 
by high-resolution melting of small amplicons. 
Clin Chem, 2004. 50(7): p. 1156-64.

\medskip
\noindent
13.	Bernard, P.S., et al., Homogeneous multiplex genotyping of 
hemochromatosis mutations with fluorescent hybridization probes. 
Am J Pathol, 1998. 153(4): p. 1055-61.


\medskip
\noindent
14.	Margraf, R.L., et al., Genotyping hepatitis C virus by 
heteroduplex mobility analysis using temperature gradient capillary 
electrophoresis. J Clin Microbiol, 2004. 42(10): p. 4545-51.

\bye
