1.基本算法(摘自维基百科)：
1.1空位权值恒定模型算法
1.2通用算法
其中H(i,j)是最终的得分矩阵。F(i,j)和E(i,j)矩阵分别用来存储在两条比对序列上开辟空位延伸比对的消耗(cost)。o代表第一个空位的罚分值，e代表延伸时的罚分值。
(原文内容：In (3) and (4), o denotes the gap opening penalty, while e represents the gap extension penalty. The matrices F(i, j) and E(i, j) contain the trace of opening and extending a gap, respectively. F(i, j) stores the cost for opening a gap and extending a gap on sequence x, while E(i, j) stores the cost for opening a gap and extending a gap on sequence y)
2.举例说明：
比对序列为：A = TGTTACGG，B = GGTTGACTA
2.1确定置换矩阵和空位罚分办法
置换矩阵s(ai,bj)= {+3，ai==bj }
{-3，ai!=bj }
即碱基匹配时分数+3，不匹配时分数-3
空位罚分Wk=(k-1)+2
即第一个空位得分-2，随后得分递增减１，即连续两个空位得分-3，连续三个空位得分-4...
2.2创建矩阵并初始化
需要创建的矩阵有H(得分矩阵)，E(B序列空位延伸罚分矩阵)，F(A序列空位延伸罚分矩阵)。
在初始化时一般会在H矩阵的最左上方空位赋一个初值(8)，右侧值和下侧值为拓展空位罚分的数值。
(H矩阵初始化)
(Matrix H)
|T |G |T |T |A |C |G |G | (A)
------------------------------------------
|8 |6 |5 |4 |3 |2 |1 |0 |0 |
G |6 |x |x |x |x |x |x |x |x |
G |5 |x |x |x |x |x |x |x |x |
T |4 |x |x |x |x |x |x |x |x |
T |3 |x |x |x |x |x |x |x |x |
G |2 |x |x |x |x |x |x |x |x |
A |1 |x |x |x |x |x |x |x |x |
C |0 |x |x |x |x |x |x |x |x |
T |0 |x |x |x |x |x |x |x |x |
A |0 |x |x |x |x |x |x |x |x |
(B)
(E矩阵初始化)
(Matrix E)
|T |G |T |T |A |C |G |G | (A)
------------------------------------------
| | | | | | | | | |
G | |0 |0 |0 |0 |0 |0 |0 |0 |
G | |x |x |x |x |x |x |x |x |
T | |x |x |x |x |x |x |x |x |
T | |x |x |x |x |x |x |x |x |
G | |x |x |x |x |x |x |x |x |
A | |x |x |x |x |x |x |x |x |
C | |x |x |x |x |x |x |x |x |
T | |x |x |x |x |x |x |x |x |
A | |x |x |x |x |x |x |x |x |
(B)
(F矩阵初始化)
(Matrix F)
|T |G |T |T |A |C |G |G | (A)
------------------------------------------
| | | | | | | | | |
G | |0 |x |x |x |x |x |x |x |
G | |0 |x |x |x |x |x |x |x |
T | |0 |x |x |x |x |x |x |x |
T | |0 |x |x |x |x |x |x |x |
G | |0 |x |x |x |x |x |x |x |
A | |0 |x |x |x |x |x |x |x |
C | |0 |x |x |x |x |x |x |x |
T | |0 |x |x |x |x |x |x |x |
A | |0 |x |x |x |x |x |x |x |
(B)
2.3打分
由通用算法的计算式知：
E(0,1)=max(H(0,0)-2=8-2,E(0,0)-1=0-1)=6
E(1,1)=max(H(1,0)-2=6-2,E(1,0)-1=0-1)=4
F(1,0)=max(H(0,0)-2=8-2,F(0,0)-1=0-1)=6
F(1,1)=max(H(0,1)-2=6-2,F(0,1)-1=0-1)=4
H(1,1)=max(0,H(0,0)=8-3,E(1,1),F(1,1))=5
......
第一轮打分后：
(H矩阵)
(Matrix H)
|T |G |T |T |A |C |G |G | (A)
------------------------------------------
|8 |6 |5 |4 |3 |2 |1 |0 |0 |
G |6 |5 |9 |x |x |x |x |x |x |
G |5 |3 |x |x |x |x |x |x |x |
T |4 |x |x |x |x |x |x |x |x |
T |3 |x |x |x |x |x |x |x |x |
G |2 |x |x |x |x |x |x |x |x |
A |1 |x |x |x |x |x |x |x |x |
C |0 |x |x |x |x |x |x |x |x |
T |0 |x |x |x |x |x |x |x |x |
A |0 |x |x |x |x |x |x |x |x |
(B)
(E矩阵)
(Matrix E)
|T |G |T |T |A |C |G |G | (A)
------------------------------------------
| | | | | | | | | |
G | |0 |0 |0 |0 |0 |0 |0 |0 |
G | |6 |4 |3 |2 |1 |0 |0 |0 |
T | |5 |3 |7 |x |x |x |x |x |
T | |4 |x |x |x |x |x |x |x |
G | |3 |x |x |x |x |x |x |x |
A | |2 |x |x |x |x |x |x |x |
C | |1 |x |x |x |x |x |x |x |
T | |0 |x |x |x |x |x |x |x |
A | |0 |x |x |x |x |x |x |x |
(B)
(F矩阵)
(Matrix F)
|T |G |T |T |A |C |G |G | (A)
------------------------------------------
G |0 |6 |5 |4 |3 |2 |1 |0 |
G |0 |4 |3 |x |x |x |x |x |
T |0 |3 |2 |x |x |x |x |x |
T |0 |2 |x |x |x |x |x |x |
G |0 |1 |x |x |x |x |x |x |
A |0 |0 |x |x |x |x |x |x |
C |0 |0 |x |x |x |x |x |x |
T |0 |0 |x |x |x |x |x |x |
A |0 |0 |x |x |x |x |x |x |
(B)
继续计算直到完整填充......
(H矩阵)
(Matrix H)
|T |G |T |T |A |C |G |G | (A)
------------------------------------------
|8 |6 |5 |4 |3 |2 |1 |0 |0 |
G |6 |5 |9 |7 |6 |5 |4 |4 |3 |
G |5 |3 |8 |6 |5 |4 |3 |7 |7 |
T |4 |8 |6 |11|9 |9 |8 |7 |6 |
T |3 |7 |x |x |x |x |x |x |x |
G |2 |5 |x |x |x |x |x |x |x |
A |1 |4 |x |x |x |x |x |x |x |
C |0 |3 |x |x |x |x |x |x |x |
T |0 |3 |x |x |x |x |x |x |x |
A |0 |x |x |x |x |x |x |x |x |
(B)
(E矩阵)
(Matrix E)
|T |G |T |T |A |C |G |G | (A)
------------------------------------------
G |0 |0 |0 |0 |0 |0 |0 |0 |
G |6 |4 |3 |2 |1 |0 |0 |0 |
T |5 |3 |7 |5 |4 |3 |2 |2 |
T |4 |2 |6 |4 |3 |2 |1 |5 |
G |3 |6 |x |x |x |x |x |x |
A |2 |5 |x |x |x |x |x |x |
C |1 |4 |x |x |x |x |x |x |
T |0 |3 |x |x |x |x |x |x |
A |0 |2 |x |x |x |x |x |x |
|0 |1 |x |x |x |x |x |x |
(B)
(F矩阵)
(Matrix F)
|T |G |T |T |A |C |G |G | (A)
------------------------------------------
G |0 |6 |5 |4 |3 |2 |1 |0 |0 |
G |0 |4 |3 |7 |6 |5 |4 |3 |2 |
T |0 |3 |2 |6 |5 |4 |3 |2 |5 |
T |0 |2 |6 |5 |7 |9 |8 |7 |6 |
G |0 |1 |x |x |x |x |x |x |x |
A |0 |0 |x |x |x |x |x |x |x |
C |0 |0 |x |x |x |x |x |x |x |
T |0 |0 |x |x |x |x |x |x |x |
A |0 |0 |x |x |x |x |x |x |x |
(B)
这里不再计算，留给读者自己验证……
参考资料：
1.Optimized and Portable FPGA-Based Systolic Cell Architecture for Smith–Waterman-Based DNA Sequence Alignment
2.Wiki-en