Figure 1.
Approximations of a highly rugged fitness landscape by broad peaks and neutral regions.
The figures depict examples of highly rugged fitness landscapes where the sequence space has been projected in one dimension. (A) Sequences with fitness below some level are functionally very different to the desired function, and selection cannot act upon them. All other sequences are considered as targets. The fitness landscape is approximated by a step function: if
, then
, otherwise
. (B) Local maxima below the desired fitness threshold
are known to slow down the evolutionary random walk towards sequences that attain fitness at least
. We approximate the fitness landscape by broad peaks and neutral regions by increasing the fitness of every sequence that belongs in a mountain range with fitness below
to the maximal local maxima
below
. Note that the target set starts from the upslope of a mountain range whose peak exceeds
.
Figure 2.
Broad peak with different fitness landscapes.
For the broad peak there is a specific sequence, and all sequences that are within Hamming distance are part of the target set. The fitness landscape is flat outside the broad peak. (A) If the width of the broad peak is
, then the expected discovery time is exponential in sequence length,
. (B) If the width of the broad peak is
, then the expected discovery time is polynomial in sequence length,
. (C) Numerical calculations for broad peak fitness landscapes. We observe exponential expected discovery time for
and
, whereas polynomial expected discovery time for
.
Table 1.
Numerical data for discovery time in flat fitness landscapes.
Figure 3.
The search for randomly, uniformly distributed targets in sequence space.
(A) The target set consists of random sequences; each one of them is surrounded by a broad peak of width up to
. The figure shows a pictorial illustration where the
-dimensional sequence space is projected onto two dimensions. From a randomly chosen starting sequence outside the target set, the expected discovery time is at least
, which can be exponential in
. (B) Computer simulations showing the average discovery time of
,
, and
targets, with
. We observe exponential dependency on
. The discovery time is averaged over 200 runs. (C) Success probability estimated as the fraction of the 200 searches that succeed in finding one of the target sequences within
generations. The success probability drops exponentially with
. (D) Success probability as a function of time for
and
. (E) Discovery time for a large number of randomly generated target sequences. Either
or
sequences were generated. For
and
the target set consists of balls of Hamming distance
and
(respectively) around each sequence. The figure shows the average discovery time of 100 runs. As expected we observe that the discovery time grows exponentially with sequence length,
.
Figure 4.
Gene duplication (or possibly some other process) generates a steady stream of starting sequences that are a constant number of mutations away from the target. Many searches drift away from the target, but some will succeed in polynomially many steps. We prove that
searches ensure that with high probability some search succeed in polynomially many steps.