Sequence Comparison.pdf

160 8 Scoring Matrices
k variables with m constraints to a problem in k + m variable with no constraints.
The new objective function is a linear combination of the original objective function
and the m constraints in which the coefficient of each constraint is a scalar variable
called the Lagrange multiplier.
Here we introduce 20 Lagrange multipliers α i for the constraints in (8.13), 19
Lagrange multipliers β j for the first 19 constraints in (8.14), and additional Lagrange
multiplier γ for the constraint in (8.15). To simplify our description, we define β 20 =
0. Consider the Lagrangian
F ((Q ij ),(α i ),(β j ),γ)
= D(Q ij )+∑α i
(P i −∑
i
j
[ ( )
q ij
+γ ∑q ij ln
ij
P i P
j
′
Q ij
)
+∑
−∑Q ij ln
ij
)
β j
(P j ′ −∑Q ij
j
i
( )]
Q ij
P i P ′ j
. (8.16)
Setting the partial derivative of the Lagrangian F with respect to each of the Q ij
equal to 0, we obtain that
( ) ( ( ( ) ))
Qij
Q ij
ln + 1 − α i + β j + γ ln
q ij P i P
j
′ + 1 = 0. (8.17)
The multidimensional Newtonian method may be applied to equations (8.13) –
(8.15) and (8.17) to obtain the unique optimal solution (Q ij ).
After (Q ij ) is found, we calculate the associated scoring matrix (S ij ) as
( )
S ij = 1 λ ln Q ij
P i P
j
′ ,
which has the same λ as the original scoring matrix (s ij ). The constraint (8.17) may
be rewritten as
Q ij = e (αi−1)/(1−γ) e (β j+γ)/(1−γ) q 1/(1−γ) (
ij Pi P j
′ ) −γ/(1−γ) .
Table 8.4 PAM substitution scores (bits) calculated from (8.18) in the uniform model.
PAM distance Match score Mismatch score Information per position
5 1.928 -3.946 1.64
30 1.588 -1.593 0.80
47 1.376 -1.096 0.51
70 1.119 -0.715 0.28
120 0.677 -0.322 0.08