DNAModel

In a quantum mechanical context, each base in a base pair can be treated as a system of particles (protons, neutrons, and electrons). The wave function for this system would be a solution to the Schrödinger equation, and would give the probability distribution for each of these particles.

The base pair could then be modeled as a composite system of two such base systems, and you would likely need to account for the hydrogen bonding interaction between the bases in your model.

Here's how you might approach it mathematically:

• For each atom in the base, you can write down the wavefunction of its electrons in the atomic orbitals.
• For the entire base (made up of several atoms), you can construct the molecular orbital by considering the combination of these atomic orbitals.
• For a base pair, which consists of two bases, you would consider the interaction between the two bases.

This approach involves approximations (like treating each atom or base separately before considering their interactions), because the full many-body problem of all the particles in the base pair is extremely complex and can't be solved exactly.

Note that constructing a quantum mechanical model of a DNA base pair from first principles is a complex task, and even with powerful computational resources, it's usually done using various approximation methods.

Asolution to non-perturbative Yang-Mills quantization may bring new insights into other related questions as well.

First Draft of a Stochastic Model

While DNA replication is not traditionally modeled as a stochastic dynamical system, one can indeed conceptualize it as such. Let's set up an abstract mathematical representation of this process.

The states in our system could be defined as each possible base pair configuration for a given position on the DNA strand: (A-T), (T-A), (C-G), and (G-C). Note that while physically A always pairs with T and C always pairs with G, we are considering mismatches as possible states for mathematical completeness and to account for potential mutations.

The replication process could then be viewed as a transition from one state (the original DNA strand) to another state (the new DNA strand). The transitions between states could be modeled by a transition matrix that represents the probabilities of moving from one base pair to another during replication. This is where the stochastic nature of the system comes in.

If we denote the states as $A$, $T$, $C$, $G$ for simplicity (understanding that each letter also implies its complementary base due to the base pairing rule), a hypothetical transition matrix might look something like:

	$A$	$T$	$C$	$G$
$A$	0.99	0.01	0	0
$T$	0.01	0.99	0	0
$C$	0	0	0.99	0.01
$G$	0	0	0.01	0.99

This matrix states that if we are in state $A$ (or $T$, because of complementarity), there is a 99% chance we'll correctly replicate to state $A$ (or $T$) again, but a 1% chance of a mutation occurring to state $T$ (or $A$). The same logic applies to states $C$ and $G$. This matrix represents a simple model where mutations can only happen as transitions between complementary pairs, but in reality, the situation can be much more complex.

Remember that this is a very abstract, simplified model of DNA replication and doesn't capture many aspects of the real, biological process. However, it can serve as a starting point to think about DNA replication as a stochastic dynamical system. Modifications and additions would be necessary to represent more realistic scenarios or to model more specific situations, such as the influence of external factors on the mutation rate.

TODO: Relate this to the very similiar sequence derived from the Riemann zeta function by Dr Broughan