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

\bigskip

\centerline{\bf Mathematical model for melting curves and heteroduplex content}

\bigskip
In Figure 1, we show high-resolution melting curves 
of the DNA amplicon from three genotypes of a SNP for which
the homozygous mutation is indistinguishable from that 
of the wild type, due to nearest-neighbor thermodynamic
symmetry of the mutation. (This says that the bases
immediately surrounding the mutation are identical
when the strands are interchanged, e.g., TCA/AGT $\leftrightarrow$ TGA/ACT.)
Our goal is to add the right fraction of wild-type
DNA to each genotype before PCR, so that after amplification,
melting, and reannealing, the mixture with the
homozygous mutant sample will develop heteroduplex
content but the mixture with the wild-type sample will
not, making the curves distinguishable. The mixture with
the heterozygous sample will have its heteroduplex content
reduced from its natural 50 \% value, so we must be careful
that its curve remains distinct from the heteroduplex-enhanced
homozygous sample.

\bigskip

Therefore, in this section, we develop a model for the melting
curve of a mixture of genotypes in terms of the melting
curves of the consituent duplex types and the mixture
fractions.
In the case above when the homozygous mutant and wild-type
curves are indistinguishable,
we show that the difference among curves of different
genotypes depends solely on the heteroduplex content
of the mixture. 
In addition, the temperature of maximum separation
does not change with heteroduplex content.
This is in spite of the maybe surprising fact that the 
two heteroduplex species have different thermodynamic melting
behavior, even with respect to nearest-neighbor approximation.

\bigskip
Finally, we model and optimize the
separation of heteroduplex contents of the three
genotypes in terms of mixture fraction.

\bigskip
We begin by listing what kind and fraction
of duplexes are present 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.
The DNA mixtures are amplified by PCR, dissociated, 
then rapidly cooled so that complementary
and nearly complementary strands form duplexes
independently, i.e., solely in fraction to their
fractions.
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 1a lists the resulting
duplex fractions.

\bigskip
After denaturation and cooling, fractions of four duplexes are
obtained by multiplying the corresponding strand 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.
Assuming fluorescence is additive,
the normalized melting curves for each of the genotypes
are described by

$$W(T)= 1 F_1(T),$$
$$M(T,x) = x^2 F_1(T) + x(1-x) F_2(T) + x(1-x) F_3(T) + (1-x)^2 F_4(T),$$
$$H(T,x) =  {(1+x)^2 \over 4} F_1(T) + { 1-x^2 \over 4} F_2(T) +
{ 1-x^2 \over 4} F_3(T) + {(1-x)^2 \over 4} F_4(T) .$$

\bigskip

Our goal is to maximize our ability to distinguish these
three melting curves, as measured by the minimum separation between
any two of them, defined as the
maximum absolute value of their difference:

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

For this reason we consider
$$W(T,x)- M(T,x) = 
(1-x^2) F_1(T) - x(1-x) F_2(T) - x(1-x) F_3(T) - (1-x)^2 F_4(T)$$

and

$$W(T,x) - H(T,x) =  (1-{(1+x)^2 \over 4} ) F_1(T) - { 1-x^2 \over 4} F_2(T) -
{ 1-x^2 \over 4} F_3(T) - {(1-x)^2 \over 4} F_4(T) .$$

We will determine $H(T,x) - M(T,x)$ as the difference of the differences
above, after simplifying.

\bigskip

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.
Therefore we assume $F_1(T)=F_4(T)$,
as predicted by the nearest-neighbor approximation for
SNPs with the symmetry described in the previous section.
This symmetry does not make $F_2$ and $F_3$ agree, but
this is not required for our derivation.
Using the subscript $\ _=$
to indicate melting curves referring to
this case, we replace $F_4$ by $F_1$ and combine terms to get

$$W_=(T,x)- M_= (T,x) = 
(2x-2x^2) F_1(T) - x(1-x) F_2(T) - x(1-x) F_3(T)$$

and

$$W_=(T,x) - H_=(T,x) =  ({ 1-x^2  \over 2} F_1(T) - { 1-x^2 \over 4} F_2(T) -
{ 1-x^2 \over 4} F_3(T) .$$

\bigskip
Finally, noticing that the coefficient of $F_1$ in both expressions is
minus twice the common coefficients of $F_2$ and $F_3$, 
we divide it equally between $F_1$ and $F_4$ again to arrive at

$$M_=(T,x)- W_=(T,x) = m(x)F(T)$$

$$H_=(T,x)- W_=(T,x) = h(x)F(T)$$

and therefore

$$H_=(T,x)- M_=(T,x) = (h(x)-m(x))F(T),$$

where

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

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

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

Since there are two types of heteroduplex, $m(x)$
and $h(x)$ represent the total heteroduplex content of
the mutant homozygous and heterozygous plus wild-type mixtures,
respectively, and along with $h(x)-m(x)$, the
difference in heteroduplex content between any
two mixtures.
The graphs of the absolute values of the three heteroduplex
content difference functions are given in
Figure 2a, annotated with key features derived below.
The graphs of the theoretical duplex melting curves $F_j(T)$
and the corresponding $F(T)$ are given in Figure 2b.

\bigskip
These expressions have two important consequences.
First, they show that in the case of interest,
the pointwise separation of the melting curves
of different genotypes
is proportional solely to 
the difference in heteroduplex content of the mixtures.
Second, they uncouple the $x$ (fraction of added DNA) and $T$
(temperature) dependence of the differences among fluorescence
curves of different genotypes. This allows us to optimize
the fraction added 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

Symbolically, our original genotype separation problem reduces to
$$ \max_{x \in [0,1]} \min \{ m(x)  , h(x)  , | h(x) - m(x) | \}  F ,$$
where
$$F= \max_{T} F(T)$$
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 $h(0)={1 \over 2}$.

\bigskip
What remains is to optimize the fraction of added DNA $x$ 
to maximize
$$\min \{ m(x)  , h(x)  , | h(x) - m(x) | \} .$$ 
In Figure 2a, this corresponds to finding 
the highest point on the lowest of the graphs of $m$, $h$,
and $|m-h|$.
Calculus confirms 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. To find these points, we first decide which
of the functions has the minimum value for each value
of $x$. 
To discover which is the smaller of $h(x)$ and $m(x)$
as well as to simplify $|h(x)-m(x)|$, we locate
the points where $m(x)=h(x)$ or $h(x)-m(x)=0$.
Using 
$h(x)-m(x)={3 x^2 -4x + 1 \over 2}= {1 \over 2}(3x-1)(x-1)$, 
we find that
for $x<{1 \over 3}$,
$m(x)<h(x)$ and $|h(x)-m(x)|=h(x)-m(x)$.
Further comparing $m(x)$ and $h(x)-m(x)$, we find that
they are equal in this interval when $h(x)-2m(x)=
{7 \over 2}x^2 -4x +{1 \over 2}={1 \over 2}(7x-1)(x-1)
=0$, or $x={1 \over 7}$.
So for $0<x<{1 \over 7}$,
$m(x)<h(x)-m(x)<h(x)$, and for ${1 \over 7} < x < {1 \over 3}$
$h(x)-m(x)<m(x)<h(x)$. For ${1 \over 3}<x<1$, the
order becomes $h(x)-m(x)<h(x)<m(x)$. The only place the
minimum of these functions has a horizontal tangent is
halfway between its roots ${1 \over 3}$ and $1$, 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.
For this value, the separation between the wild-type 
melting curve and the other two melting curves
will be even greater.

\bigskip

The only places the graph of the minimum of
the three functions does not have a tangent are where it
changes form, i.e., at $x={1 \over 7}$ and $x={1 \over 3}$.
Comparing the values at these points, we find the
optimal fraction of added DNA occurs at 
$x=x_*={1 \over 7}$, as indicated in Figure 2a.
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 
fraction of wild-type DNA to add.
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 fraction added and temperature problems.
In this case, the optimal added 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 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.

\bye
