APPENDIX B - Letter for fourth discussion

Simon Ruijsenaars writes, “Although I have no comments as yet concerning the mathematical modeling of solitary excitations in DNA, the topological concepts you present in the last communication remind me of Wigner’s comments concerning the use of mathematics to understand nature. I am looking forward to hearing more about how topology enters into understanding the interactions between different types of biomacromolecules”.

Henry Sobell answers, “Yes, I have been aware of Wigner's comments. The interesting thing is that much of our initial understanding came from the use of the Corey-Pauling-Kolten (CPK) space filling molecular models of DNA fragments to study its interactions with intercalators. These models incorporated the structural information we obtained from our crystallographic studies and then served as analog computers, guiding us to carry out the energy minimization studies that followed. The latter studies allowed us to directly verify the nontopological nature of premeltons arising within B- and A- DNA.

[Simon Ruijsenaars is a mathematical physicist at the Center for Mathematics and Computer Science in the Netherlands].

Thomas Broker writes, “It was great to hear from you. I look forward to reading the discussion, and learning more about the subject.

Henry Sobell answers, “Your discovery by electron microscopy of a four-way junction connecting T4 bacteriophage DNA molecules undergoing genetic recombination, and your use of (black and white) velcro strands to demonstrate the movement of homologous DNA molecules upon each other, makes you a pioneer in this subject! This is because the junction (called the Holliday junction) is -- in reality -- an example of a higher energy “discrete breather”, whose breather motions facilitate the slippage of DNA chains on one another! We will describe these motions in a subsequent communication.

[Thomas Broker studies the human papillomavirus -- a leading cause of cervical cancer –- as Professor of Biochemistry and Molecular Biology, University of Alabama, Birmingham, Alabama].

So – shall we now continue?

How does the B to A structural phase transition take place?

In the presence of suitable thermodynamic conditions (i.e., those that create a bias that favors the formation of A-DNA), kink and antikink within premeltons in B-DNA structure (B-B premeltons) begin to move apart to form larger and larger core regions -- whose centers modulate into A-DNA structure. This necessarily involves the formation of B-A premeltons and A-B premeltons, which act as phase boundaries that continue to move apart, leaving A-DNA to form within. Finally, long regions of A-DNA appear, containing A-A premeltons embedded within.

Such a mechanism is reversible and illustrates how a bifurcation within the central core regions of B-B or A-A premeltons (present within B- or A-DNA) can give rise to this structural phase transition.

[NOTE: By a bifurcation, we mean an event that takes place at a branch point in a pathway to give rise to two different outcomes. Although the source of the nonlinearity (i.e., that determines the pathway) remains the same, the decision as to which pathway to take at the branch point is influenced by a bias. In the case of the B to A transition originating within the centers of premeltons, prevailing thermodynamic conditions provide the bias].

Now we come to a key question: What is the nature of the breather motions present in B-B (or A-A) premeltons?

To understand this, we will begin by examining an animation of the beta-structural element alternating between its lowest and highest energy states, “pinned” and “unpinned” by irehdiamine and ethidium. Irehdiamine (a steroidal diamine – shaped like a banana) partially intercalates into DNA, while ethidium (a phenanthridinium derivative -- shaped like a plate) completely intercalates into DNA.

From where does the energy come to effect such a dynamic interconversion?

As seen in the animation above, the movement of the kink and antikink is tightly coupled with the appearance of the lowest and highest energy states of the central beta-structural element. The extremes in these two conformational states limit the excursion of kink and antikink, causing the kink and antikink to remain together as a “kink-antikink bound state” (or equivalently -- in a lattice structure –- a “discrete breather”).

[NOTE: Premeltons containing longer regions of beta-DNA in their centers have higher energies, and it is therefore not unexpected that they will exhibit more complex breather motions. We will not discuss these in any detail here].

Henry Sobell

Alwyn Scott