Evolutionary Path in Shape Space

Our report defines an "evolutionary path in shape space". Its definition is necessarily terse in that context. A few additional remarks might be in order to appreciate its full meaning. We provide them here for the interested reader.

Discussion of the transitions

While secondary structure graphs are easy to understand visually, it is difficult to visually parse the exact difference between any two of them. This is made easier by representing secondary structures as linear strings of matching parentheses. Each matching pair of parentheses represents a base pair between the corresponding positions along the sequence, and a dot stands for an unpaired position. For example, here is a short sequence with a simple hairpin loop as its secondary structure: 

CGAUGCGCGCAUCGGAU
(((((....)))))...
Positions 1,2,3,4,5 are paired with positions 15, 14, 13, 12, 11, 10, respectively.


Transition a (7 to 8)

A fitness improving continuous transition (loss of one base pair, marked by a pair of asterisks below) 

                *                       *
7  .(((((((((((((.......(((....)))......)))))).....(((((.......))))))))))))....
8  .((((((((((((........(((....))).......))))).....(((((.......))))))))))))....
                *                       *

Transition b (8 to 9)

A fitness improving continuous transition (loss of one base pair) 

          *                                                         *
8  .((((((((((((........(((....))).......))))).....(((((.......))))))))))))....
9  .((((((.(((((........(((....))).......))))).....(((((.......))))).))))))....
          *                                                         *

Transition c (10 to 11)

A fitness neutral discontinuous transition of the simple shift type: 

                               ***
10 .(((((..(((((........(((....))).......))))).....(((((.......)))))..)))))....
11 .(((((..(((((........(((...)))........))))).....(((((.......)))))..)))))....
                              ***

Transition d (13 to 14)

This is a discontinous transition of the generalized shift type (a double flip with simple shift; see figure 3 of our report) creating the multiloop. 

13 (((((((.(((((........(((...)))........))))).....(((((.......))))).)))))))...	
               aaa      AAA   BBB      bbb                 
14 (((((((.(((((((.......)))..(((......))))))).....(((((.......))))).)))))))...

In the transition from shape 13 to shape 14 the positions labelled "AAA" and paired with "BBB" in 13 flip around to pair with downstream positions "aaa" (at the same time "AAA" shifts by one position to the 5' end). Simultaneously the paired positions "BBB" flip to pair upstream to "bbb".

Transition e (18 to 19)

A discontinuous transition of the simple shift type. 

               **
18 (((((((.(((.((((.....)))).((((......))))))).....(((((.......))))).)))))))...
19 (((((((.(((((.((.....)))).((((......))))))).....(((((.......))))).)))))))...
              **

Transition f (36 to 38 via 37)

A discontinuous transition of the generalized shift type (a flip).

The flip is best visible in the comparison between 36 and 38. Notice the short lived intermediate 37 (whose fitness is decreased with respect to both 36 and 38).

In 36 "XXX" pairs with "xxx", and "AAAAA" with "aaaaa". In 38, after the flip, "AAAAA" pairs with a new region "bbbbb" comprising the former "xxx" which gave up its pairing to "XXX". The * indicates a continuous transition, adding one base pair, hitchhiking on this discontinuous change. 

           XXX               AAAAA    aaaaaxxx
36 ((((((..(((((((......)))).(((((....)))))))).....(((((.......)))))..))))))...
37 ((((((....(((((......)))))((((((....)).)))).....(((((.......)))))..))))))...
38 ((((((....(((((......)))))(((((.......))))).....(((((.......)))))..))))))... 
             *              * AAAAA       bbbbb

Transition g (39 to 41 via 40)

A discontinuous transition of the simple shift type. Notice again the interesting short lived intermediate 40. 

              ****
39 ((((((.....((((......)))).(((((.......))))).....(((((.......)))))..))))))...
40 ((((((...((((((......)))).(((((.......))))).....(((((.......)))))))))))))...
41 ((((((....((((.......)))).(((((.......))))).....(((((.......)))))..))))))...
             ****

Transition h (41 to 42)

A discontinuous transition of the simple shift type. 

             ****
41 ((((((....((((.......)))).(((((.......))))).....(((((.......)))))..))))))...
42 ((((((...((((........)))).(((((.......))))).....(((((.......)))))..))))))...
            ****

Transition i (42 to 44 via 43)

A discontinuous transition of the simple shift type. The shifting helix half is rather long, and the shift happens in two stages with a very short lived intermediate. 

	                                                              ******
42 ((((((...((((........)))).(((((.......))))).....(((((.......)))))..))))))...
43 ((((((...((((........)))).(((((.......))))).....(((((.......))))).)).))))...
44 ((((((...((((........)))).(((((.......))))).....(((((.......))))).))))))....
                                                                     ******

Atavism event j

This is not a transition, but a quite interesting effect. Sequences with shapes 29 and 30, currently on the evolutionary path, revert in a single point mutation to sequences whose shape is 9, an ancestral shape on the path. By placing a few suitable mutations some polymers may "remember" their past.