Feynman diagrams, RNA folding, and the transition polynomial

Yongwu RongDepartment of Mathematics

George Washington University

RNA in Biology, Bioengineering and Nanotechnology

IMA, October 29-November 2, 2007

Feynman Diagrams in Physics

Photonwavy line

Positronline, arrow to the left straight

Electronstraight line, arrow to the right

ParticleDescriptionImage

Feynman Diagrams in Physics

An electron and a positron meet and annihilate (disappear),producing a photon

A photon produces an electron and a positron (an electron-positron pair)

A positron absorbs a photon

A positron emits a photon

An electron absorbs a photon

An electron emits a photon

Feynman Diagrams in Physics

Feynman Diagram in Mathematics

• Topology: Classify spaces in any dimension

Dim = 1.

Dim = 2. plus Klein bottles etc

genus = # of handles.

Dim = 3. Recently solved by Perelman. Dim > 3, proved to be impossible to classify

(undecidable).

Feynman Diagrams in Mathematics

Knots:

Kelvin conjectured that atoms were knotted tubes of ether.

Periodic Table of Elements?Periodic Table of Elements?

Feynman Diagrams in MathematicsKnots:

Singular knots:

Chord diagram (Feynman diagrams)

Came from Vassiliev invariants (part ofQuantum Invariants)

Feynman Diagrams in Mathematics

Can add, subtract, multiply Feynman diagrams.

2

After taking a “quotient,” get “M = space of Feynman diagrams” ~ “Space of Vassiliev Invariants.”

Feynman Diagrams in Mathematics

• Conjecture. They classify all knots.• Conjecture. dim M ~ e n , as n infinity.• Can fit knots in M using “Kontsevich integral.”

e.g.

1/12 11/24

5/24 1/8 1/8 1/12

higher order terms

Feynman Diagrams in Mathematics

• If the chord don’t intersect each other, the diagram is planar.

• These diagrams generate the so-called “Temperley-Liebalgebra” in statistical mechanics.

• The number of such diagrams with n chords, i.e. dimension of “Temperley-Lieb algebra,” is the

“Catalan number.”

Example. n = 3. dim = 5.

Feynman Diagrams in Mathematics

• In general, the chords may intersect each other, and therefore yields non-planar Feyman diagrams.

• The extent of non-planarness can be measured by genus – the smallest genus of a surface it lies on.

e.g.

g = 1

Watson-Crick pair: A-U, C-Gwobble pair: G-U.

RNA Primary Structure: A C C U G U A G U A A U G A G U C U

RNA secondary structure:

RNA tertiary structure:

Feynman Diagrams in Biology

Feynman Diagrams in Biology

Secondary structures are coded by Feynman diagrams:

Secondary structure without pseudoknots:

Feynman Diagrams in Biology

Secondary structures are coded by Feynman diagrams:

Secondary structure pseudoknots:

Feynman Diagram in Biology

Basic Questions:Given primary structure of an RNA, determines its secondary structure, and tertiary structure that minimizes the “energy.”

Often, energy = free energy from pairing +stacking energy.

Feynman Diagram in Biology

Partition function

summed over all possible Feynman diagrams D.(Compare to Kontsevitch integral).

This gives the probability distribution of states. The one with minimum volume has the highest probability.

∑−

=D

DEKTeZ

)(1

∑∑ ==−

DD

DEKT Dpe

Z)(

11

)(1

H. Orland, M. Pillsbury, A. Taylor, G. Vernizzi, A. Zee reformulated the RNA folding problem as an N × N matrix field theory. The terms in the partition function are classifiedaccording to their topological character. In particular, the genus of a Feynman diagram plays an important role.

For more details, see talk by Henri Orland tomorrow.

Our work:

1. Classified all genus one Feynman diagrams.2. Introduced a “transition polynomial” which contains some

useful topological information such as genus.

Feynman Diagram in Biology

Pillsbury, Orland, Zee (2005) classified all genus one irredubilepseudoknot using “Matrix Field Theory,” eight of them total.

a

b

c

And plus trivial chords, All have genus = 1

Proposition. (Luse – R.) These are all Feynman diagrams with genus = 1.

g = 1

The “transition polynomial” for 4-valent planar-graphs was introduced by F. Jaeger. Its description is as follows.•Given a 4-regular planar graph G, color its complementary regions B or W so that the infinite region is W.•At each vertex v, there are three possible “transitions” at v.

1)()()()(),,,,( −∑= scsxswsb

states

GQ τγβατγβα

But 4-regular planar graphs can be translated to Feynman diagrams. e.g.

Try to translate “transitions” to Feynman diagrams. Need “signed Feynman diagrams.”

We define a transition polynomial for signed Feynman diagrams.

1)()()()(),,,,( −∑= scsxswsb

states

wzyxwzyxDQ

Properties. Let D be a Feynman diagram on n chords. Then

(1) Q (D, x, y, z, 1) = (x + y + z)n

(2) Q (D # D’) = Q(D) Q(D’)(3) If each chord of D has a positive sign, then

Q(D, 0, 1, 0, w) = wc-1, and genus (D) = ½ (n + 1 – c).

Further properties to be investigated:- Relation with graph polynomials (chromatic, Tutte, Penrose…)- Relations with knot polynomials (Jones, Kauffman bracket …)- Geometric meaning? Physical / biological meaning?

The End