This is the sort of question that people consider when teaching intro topology courses. For example at the Ross Mathematics Camp this summer, Jim Fowler gave this example (on slide 3) of a Moebius band appearing in biology:
A yeast cell has some sort of life cycle (grow, sleep, reproduce, or something); the life cycle can be thought of as the circle $S^1$. Apparently, a colony of yeast cells tends to synchronize its life cycles. If you have two yeast colonies $A$ and $B$, then they will be separately synchronized, but the two colonies will be out of phase in general. If you combine the two colonies into one big colony, then they will gradually re-synchronize to be in phase. The question is, how is this new phase related to the old ones?
Suppose that the resulting phase depends only on the two input phases (this is particularly reasonable if the colonies $A$ and $B$ are the same size, etc.). That is to say, it is determined by a function $f: S^1 \times S^1 \rightarrow S^1$ of the input phases. If $A$ and $B$ happen to be in phase to start, then of course we'd expect the phase to remain the same: $f(x,x) = x$. We're not allowing anything to distinguish $A$ from $B$, so $f$ ought to be symmetric, i.e. $f(a,b) = f(b,a)$, so it actually defines an $S^1$-valued function on the Moebius band $M$. Reasonably, $f$ should be continuous.
So what we're asking for is a retract from $M$ to its bounding circle. Of course, this is absurd. So there is no such function.
I'm not sure what the resolution is, and unfortunately don't have a reference to the original work. Adding in parameters, making things probabilistic... the obvious approaches don't easily resolve the underlying topological issue!