如何找到最佳字符串内容，使字符计数向量与其参考字符串的 MSE 最小化

我有以下参考序列：

ref_seq      <- "MGHQQLYWSHPRKFGQGSRSCRVTSNRHGLIRKYGLNMSRQSFR"

和这个种子模式字符串：

seed_pattern <- "FKDHKHIDVKDRHRTRHLAK??????????"

该模式中有 10 个通配符 (?)。

鉴于此功能：

aa_count_normalized <- function(x) {
  AADict <- c(
    "A", "R", "N", "D", "C", "E", "Q", "G", "H",
    "I", "L", "K", "M", "F", "P", "S", "T", "W", "Y", "V"
  )
  AAC <- summary(factor(strsplit(x, split = "")[[1]], levels = AADict),
                 maxsum = 21
  ) / nchar(x)
  AAC
}

aa_count <- function(x) {
  AADict <- c(
    "A", "R", "N", "D", "C", "E", "Q", "G", "H",
    "I", "L", "K", "M", "F", "P", "S", "T", "W", "Y", "V"
  )
  AAC <- summary(factor(strsplit(x, split = "")[[1]], levels = AADict),
                 maxsum = 21
  ) 
  AAC
}

我可以得到 ：

# we need to normalize refseq_aa content with respect
# to the length of seed_pattern, to accommodate the length
# difference between the two.
> refseq_aa_content <- aa_count_normalized(ref_seq) * nchar(seed_pattern) 
> refseq_aa_content
        A         R         N         D         C         E 
0.0000000 4.7727273 1.3636364 0.0000000 0.6818182 0.0000000 
        Q         G         H         I         L         K 
2.7272727 3.4090909 2.0454545 0.6818182 2.0454545 1.3636364 
        M         F         P         S         T         W 
1.3636364 1.3636364 0.6818182 4.0909091 0.6818182 0.6818182 
        Y         V 
1.3636364 0.6818182  

我想做的是更换通配符seed pattern- 同时保持非通配符不变 - 残差组合取自：

 AADict <- c(
        "A", "R", "N", "D", "C", "E", "Q", "G", "H",
        "I", "L", "K", "M", "F", "P", "S", "T", "W", "Y", "V"
      ) 

这样种子序列的最终氨基酸计数的均方误差（MSE）和归一化参考序列计数被最小化。

使用此 MSE 函数：

mse <- function (ref, new_seq) {
  return(mean((ref - new_seq)^2))
}

以及最终的种子序列：

seed_final.1 <- aa_count("FKDHKHIDVKDRHRTRHLAKRQQGGGSSSY")
seed_final.2 <- aa_count("FKDHKHIDVKDRHRTRHLAKRQQQGGGSSY")
seed_final.3 <- aa_count("FKDHKHIDVKDRHRTRHLAKSSSGGGRRQQ") # onyambu's

I get

> mse(refseq_aa_content, seed_final.1 )
[1] 1.501446
> mse(refseq_aa_content, seed_final.2 )
[1] 1.63781
> mse(refseq_aa_content, seed_final.3 )
[1] 1.560537

The seed_final.1 is the 最优精确解，因为它的 MSE 最低。即10?s 将替换为：

G Q R S Y 
3 2 1 3 1 (total 10)

如何创建高效的 R 代码来返回FKDHKHIDVKDRHRTRHLAKRQQGGGSSSY作为答案。

您可以将问题建模为要最小化的整数二次问题：

sum(r^2) - 2 sum(z * r)

有约束条件：

sum(r) = k

r[i] nonegative integer

where:

• r[i]有多少个字母AADict你需要添加到seed_pattern
• z[i] = n(y)/n(x) * x[i] - y[i]
• x[i]第 i 个字母的计数AADict in ref_seq
• y[i]第 i 个字母的计数AADict in seed_pattern
• n(x)中的字符数ref_seq
• n(y)中的字符数seed_pattern
• k中通配符的数量seed_pattern

我没能在 R 中找到混合整数二次求解器（免费），所以这里是启发式使用DEoptimR:

ref_seq <- "MGHQQLYWSHPRKFGQGSRSCRVTSNRHGLIRKYGLNMSRQSFR"
seed_pattern <- "FKDHKHIDVKDRHRTRHLAK??????????"

AADict <- c(
  "A", "R", "N", "D", "C", "E", "Q", "G", "H",
  "I", "L", "K", "M", "F", "P", "S", "T", "W", "Y", "V"
)

aminoSummary <- function(x){
  
  f <- factor(strsplit(x, split = "")[[1]], levels = AADict)
  
  list(
    l = nchar(x),
    k = sum(is.na(f)),
    z = table(f)
  )
}

x <- aminoSummary(ref_seq)
y <- aminoSummary(seed_pattern)

M <- length(AADict)

res <- DEoptimR::JDEoptim(
  lower = rep(0, M),
  upper = rep(y$k, M) + 1,
  fn = function(r, z, k){
    r <- floor(r)
    sum(r * r) - 2 * sum(z * r)
  },
  constr = function(r, z, k) sum(floor(r)) - k,
  meq = 1,
  z = as.vector(x$z * y$l / x$l - y$z),
  k = y$k
)

rep(AADict, floor(res$par))
[1] "R" "Q" "Q" "G" "G" "G" "S" "S" "S" "Y"