# molecular clock question

Tom Holroyd tomh at BAMBI.CCS.FAU.EDU
Thu Sep 5 17:36:57 EST 1991

```Hi.  I'm doing a report on molecular clocks, and I thought I might find
some folks to talk to on the mol-evol list.  My background for this is
mathematical.

I've noticed that there is a lot of controversy over the "rate" of
evolution.  My mathematical training leads me to suppose that evolution
proceeds in a manner that may not even have a well defined "rate", let
alone a constant linear rate, as some have supposed.

Recent theoretical work in self-organizing systems suggests that organisms
live at the boundary of order and chaos.  It may sound prosaic, but consider
that order is represented by fixed, periodic behavior, never changing -
any creature that never changes is not likely to survive long - and chaos
is a type of randomness - creatures that are too random, too disorganized,
are also unlikely.  A more precise formulation exists, but this will do for
now.  The point being that systems (and organisms) that live on the boundary
between order and chaos have just the right amount of flexibility to cope
with new situations, and yet also function in an organized, ordered fashion.

Mathematically, this region is characterized by "intermittency" - meaning
there may be "bursts" of change, followed by periods of relative stability.
This is a mathematical basis for the punctuated equilibrium theory of
evolutionary change.  PE has a good theoretical foundation.

Systems that behave this way show power law scaling of the bursts with
respect to time.  The cute way of saying this is "an order of magnitude
more bursts, an order of magnitude less often."  This means that the bursts
of change happen on all time scales, or, that there is no characteristic
time for change in such systems.  This would correspond to the "rate"
of evolution, if it existed, but these arguments suggest that the rate,
or characteristic time, is undefined: a power law scaling would be the
more appropriate description.

Another avenue of attack is that the genetic structure of an organism
are equally probably anywhere along the genome (a perhaps not too safe
assumption, but it would only help my argument if it was false), then you
can examine each point on the genome and ask the question "would a
mutation here produce a viable creature?"  If the answer is "no",
What I'm driving at is that some points are going to need mutations
at other points first, if the creature is to survive.  In fact, there
may be a large cascade of changes necessary before a single mutation
can be viable.  When this happens, the probabilities for change
become linked, and you multiply them, producing distributions of
viable mutations with long tails, i.e. power law scaling.

The point of all this is that mathematically, I don't expect mutations
to occur at a constant, linear rate.  I don't even expect there to
be a well defined rate, except that the variability of the mean might
settle down over a time scale of several hundred million years, making
molecular clocks useful on large time scales only.  Even then, though,
one cannot rule out the possibility that a "burst" may be waiting
to happen right around the corner, or just happened a few millenia
ago, leading to a large error in the time estimate.  Also, over such
large time scales, there is a good chance of some event occurring that
totally redefines all the probabilities (i.e. it's a non-stationary
system).

The mathematics of percolation clusters and self-organized criticality
is also relevant here.  Life exists on the side of a sandpile, if you
know what that means.

A side note: how likely are totally neutral mutations?  Even if a codon
mutates to one coding for the same amino acid, the relative concentrations
of the appropriate tRNA, or interactions with DNA-binding regulatory
proteins could confer a selective advantage.

Any comments from the biologists (or the mathematicians)?

Tom Holroyd
Center for Complex Systems
Florida Atlantic University
tomh at bambi.ccs.fau.edu

P.S. I'd be happy to discuss any of the above mentioned models in detail.

```