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I saw a paper on the preprint server BioRxiv which applies techniques from quantum theory to the problem of modeling the insertions and deletions that occur in the molecular sequences of all organisms during their evolution. Here's a brief description of the problem, vaguely aimed at mathematical physicists.

Most of the mutations that occur in DNA (and hence in RNA and proteins) are changes in single nucleotides. (These are called substitutions). Something like this happens:

```
ACTGTTACGGAGTATGT
ACTGTTACGGAATATGT
*
```

These are easy to model with a Markov process on 4 states. Insertions and deletions are also quite common, but much harder to model. There is a huge number of things which might happen, but each has a tiny probability.

In the study of evolution, we're looking backwards. We can observe molecular sequences in modern organisms (the final states) but don't know the initial state. But we are pretty sure that there was a single initial state from which all the final states evolved. So one molecular sequence tells you little, but a pair can tell you something, and a bunch of them can tell you a lot.

In both genetics and evolution, a lot of studies start with a 'pairwise alignment' or a 'multiple sequence alignment'. A three-sequence alignment might look like this.

```
rat: ACTGTTACGGA---GTATGT
bat: ACTGTTACGGCCAAATATGT
cat: ATTGTTACGGCCAAATATGT
* *
```

Any insertions and deletions that may have occured are shown as gaps, thanks to a 'multiple sequence alignment program'. These are ad hoc 'engineering' solutions to the problem. One such is CLUSTAL W. The paper describing this software has over 40,000 citations. Other multiple sequence alignment programs are available. There are also statistical solutions to simplified versions of the problem. The preprint claims to be something new. In my words, not theirs, it seems to be a 'principled approximation'.

## Comments

Relevant book: Richard Durrett, Probability Models for DNA Sequence Evolution. Durrett is a master probabilist.

`Relevant book: Richard Durrett, Probability Models for DNA Sequence Evolution. Durrett is a master probabilist.`

What sort of quantum techniques did that paper apply? I guess I should read it myself. Sorry I've been so busy and distracted these days, Graham!

`What sort of quantum techniques did that paper apply? I guess I should read it myself. Sorry I've been so busy and distracted these days, Graham!`

I don't know enough quantum theory to be able to tell you what techniques are used! It's called

Perturbative formulation of general continuous-time Markov model of sequence evolution via insertions/deletions, Part I: Theoretical basis, its 99 pages long (and that's just part I). They call it theoretical, but I think it might be too applied for you. Bits are reminiscent of things you've said. A snippet with garbled formulas:`I don't know enough quantum theory to be able to tell you what techniques are used! It's called *Perturbative formulation of general continuous-time Markov model of sequence evolution via insertions/deletions, Part I: Theoretical basis*, its 99 pages long (and that's just part I). They call it theoretical, but I think it might be too applied for you. Bits are reminiscent of things you've said. A snippet with garbled formulas: > 1.3. Differences from the quantum mechanics > Although we borrowed the bra-ket notation and the concept of operators from the quantum mechanics (e.g., Dirac 1958; Messiah 1961a), there are some differences between quantum mechanics and the Markov model. For example, in the Markov model, we made the bra-probability vector ( ?? p(t) ) evolve, as in Eq.(1.1.2’), in order to clarify its correspondence with the traditional matrix equation for the conditional probabilities, Eq.(1.1.2). In contrast, in quantum mechanics, it is the ket-vector, ? (t) , that is usually made evolve. This is simply by convention and, if desired, we could reformulate the quantum mechanics so that the bra-vector will evolve. Another difference, which is conceptually more important, is that, in quantum mechanics, it is the squared absolute values of the scalar products, i ? (t) 2 ( i =1, 2,..., N ), that are interpreted as the probabilities (and thus satisfy i ? (t) 2 i=1 N S =1 ). In the Markov model, in contrast, it is the scalar products themselves, ?? p(t) i ( i =1, 2,..., N ), that give the probabilities (and thus satisfy ?? p(t) i =1 i=1 N S ). This should be related to another big difference that the time evolution in the quantum mechanics is in the pure-imaginary direction ( i?? ? ?t ? (t) = ˆH ? (t) , where ?? is the Planck constant and ˆH is the instantaneous time-evolution operator called the Hamiltonian), whereas the time evolution in the Markov model is in the real direction`