simple simulation of recurrent gene mutations

Question

Sorry if this question is too basic for the forum. I am trying to determine the number of recurrently mutated genes expected by chance given an equal likelihood of mutation for all genes. There are about 20,000 genes and I have ~400 mutations spread across ~100 individuals. I want to determine the probability of the same gene being mutated in multiple individuals. I have tried the following simulation. I'm surprised that I'm always getting the same number of recurrences. Am I making some mistake?

  var = 400
  individuals = 100
  genes = 2e4
 
  mutated_genes <- sample(rep(c(1,0), c(var,genes*individuals-var)))
   
  sample_mat <- matrix(mutated_genes, ncol = individuals, nrow = genes)
  
  trials <- 1e4
  
  recurrences <- vector(mode = "list", length = trials)
  
  for (i in seq_len(trials)){
      # print(i)
      
      new_rows <- sample(nrow(sample_mat))
      new_cols <- sample(ncol(sample_mat))
      new_mat <- sample_mat[new_rows, new_cols]
      recurrences[[i]] <- rowSums(new_mat)
  }
  
  n_recurrences <- sapply(recurrences, function(x) sum(x > 1))
  
  summary(n_recurrences)
 ```

Answer 1

If what you have is $100$ individuals each having $400$ mutations in one of $20,000$ genes, with all genes having the same probability of mutation, then for a quick analysis you don't need a simulation.

Overall, there are $400 \times100=40,000$ mutations. With $20,000$ genes, that comes out to 2 mutations per gene on the average over all $400$ individuals. If that's the number you were always coming up with, your simulations were making sense. There are lots of fun things you can play with in terms of the distributions of mutations among individuals and among genes in your set of $400$ samples, but the basic point is that in this scenario you should be surprised if you don't find many genes mutated in more than 1 sample.

In practice, as I'm sure you know, things aren't that simple. For the benefit of others reading this, here are some of the issues:

If you're starting with tumor samples, you are more likely to find cancer driver genes mutated because the tissues from which you are sampling have already been selected for being cancerous. For example the TP53 gene, the "guardian of the genome," is very frequently mutated in tumors.

Even beyond genes specifically driving tumor development, different genes have different probabilities of being mutated. That depends on gene size, gene transcriptional activity, and the timing of gene replication during the cell cycle. Lawrence et al. analyzed those situations in detail, working out a way to correct for those influences to determine whether particular genes are mutated at a higher rate than chance in a set of tumor samples.