Bicoloured Motzkin Paths: A Model for DNA

G. K. Iliev, A. Saguia, S. G. Whittington Chemical Physics Theory Group Department of Chemistry University of Toronto

Problem • Polymers play an important role in a wide variety of systems – Complex system arising from a simple repeated structure – Biologically important • DNA, RNA, proteins

High T

Low T

• State of the system – Use a simplified mathematical model to describe the physical system – Interested in the large scale, thermodynamic, limit for such systems

DNA • Made up of 2 strands of 4 repeating nitrogen bases (monomers) that occur in pairs (A:T, C:G) – The two strands are complementary, and the nitrogen bases interact through hydrogen bonding • Adenine=Thymine – share two H-bonds • Cytosine≡Guanine – share three H-bonds

• In DNA replication, the two strands unzip, and the complementary strand is added onto the existing template • Recent experiments allow for the micromanipulation of individual monomers – Using optical tweezers, we can study the force required to unzip duplex DNA

Modeling • The use of self-avoiding walks has been a staple in modeling physical polymers – Pros: Capture much of the physics in the problem – Cons: Mathematically cumbersome, even in 2-D

• Solution: Simplify the model – Random, reverse-restricted or directed walks – Directed walks have no steps in a particular direction, allowing a possible factorization

Directed Walk Models • Impenetrable Dyck and Motzkin Paths are the two main models of interest • Dyck paths are made up of two directed steps – NE (1,1) and SE (1, -1) steps (no westward steps), with no steps below the line y=0

• Motzkin paths have an additional E (1, 0) step

Counting with Pictures • Use the ‘directedness’ of our models to exploit the factorization of walks – Introduce a function, D(z), which defined as: D( z ) = ∑ cn z n = 1 + z 2 + 2 z 4 + 5 z 6 + 14 z 8 + 42 z10 + 132 z12 + ... n

• Note that graphically, every Dyck paths can be written as:

• Examples:

Generating Functions • For Dyck paths, we can find the g.f. as: 1− 1− 4z 2 2 4 6 8 10 12 D( z ) = = 1 + z + 2 z + 5 z + 14 z + 42 z + 132 z + ... 2 2z

• Following a similar factorization argument for Motzkin paths, we find the g.f. M(z): 1 − z − 1 − 2 z − 3z 2 2 3 4 5 6 7 M ( z) = = 1 + z + 2 z + 4 z + 9 z + 21 z + 51 z + 127 z + ... 2 2z

• The square-root singularity in the g.f.s describe the asymptotic growth of the number of walks 1 ⇒ 2 n walks for large n 2 1 For MP : z cM = ⇒ 3 n walks for large n 3

For DP : z cD =

Modeling Adsorption • Introduce a preferential hyperplane (say y=0 in Z2), where vertices in this line observe an energetic interaction – Count the interactions via x – Write down the factorization, this time keeping track of interactions as well, to get D(x, z) and M(x, z) D ( x, z ) =

2 2 − x + x 1− 4z 2

= 1 + xz 2 + ( x + x 2 ) z 4 + (2 x + 2 x 2 + x 3 ) z 6 + ...

Adsorption • Resulting g.f. has an additional singularity – zero of the denominator D ( x, z ) =

2 2 − x + x 1− 4z 2

⇒

1 z1 = , z 2 = 2

x −1 x

• The new singularity, z2, depends on the interaction parameter x – Describes how the behaviour of the walk, with respect to the surface, as we vary x – We can think of x as a Boltzmann factor [x~exp(1/T)] – For x=2, z1=z2 Æ critical point (phase transition) – Check by looking at the mean number of vertices in the zc ) surface as T changes Æ ∂(∂−(log 1 T)

Adsorption (cont’d) Low T

• Note that for high T, the mean number of vertices is 0 – Walk is desorbed, and not interacting with the surface

• Decreasing T past a critical value, we observe a non-zero fraction of vertices in the surface – Walk is adsorbed, and for TÆ0, we observe that ½ of the vertices lie in the surface Tc

• Feature of Dyck paths

High T

•

Analysis of Motzkin paths with interactions yield the same qualitative results, with few quantitative differences – Location of the phase transition – For TÆ0, we see that all vertices lie in the surface

Modeling DNA • We can think of the duplex DNA as two directed walks made up entirely of N (0, 1) and E (1, 0) steps Motzkin mapping

O – distance between vertices increases C – distance between vertices decreases H – distance between vertices remains unchanged V – distance between vertices remains unchanged

• If we think of both strands as a single ‘walk’, then there are 4 possible ‘moves’ • This suggests a possible mapping of this problem onto a Motzkin path model that has 4 steps

Motzkin ‘DNA’ • We can use our mapping to transform the previous example into our new model 9

7

Æ

11

Æ

10

8 6

12

Æ

9

Æ

5 3

4

2

3 3

2 1

9 4

8 5

6

7

10

12 11

1

• Notice that vertices that are in contact in the DNA model correspond to vertices in the surface in the Motzkin model – As such, zipped/unzipped DNA corresponds to adsorbed/desorbed Motzkin paths

Bicoloured Motzkin Paths • Varying the interaction parameter (1/T) in our model simulates a changing interaction between contacts in the DNA model – If the parameter is small (high T), the walk is desorbed, and the DNA strand is unzipped (with no force) – For large values of the parameter (low T), we find that it is energetically favourable for the walk to be adsorbed, and this results in a zipped DNA strand

• These walks also lend themselves to an elegant factorization, yielding MBC(x, z) as: M BC ( x, z ) =

2 1 , where z1 = , z 2 = z 2 ( x) 4 2 − 2 xz − x + x 1 − 4 z

Homopolymer Bicoloured Motzkin Paths • This model can be used to identify the critical value of the interaction between pairs – However, it is inadequate to study the force problem

• Modify our walk models to include a tensile force – Relax the constraint that the last vertex must lie in the surface – Resulting ‘tail’ models the elongational force – Introduce a new variable, y, which is conjugate to the height of the final vertex above the surface h

– Including this in our factorization, we can get: 4z T ( x, z ) = M BC (2 − 2 xz − x + x 1 − 4 z )(2 z − y + 2 yz + y 1 − 4 z )

Unzipping the DNA • A change in the dominant singularity signifies a critical point – We found the adsorption/desorption transition by looking where z1=z2(x) and solving for xc – To find the critical value of the force that causes the adsorbed walk to desorb, we look at z2(x)=z3(y) • Remember x=exp(-ε/T) and we can write y=exp(f/T) • This allows us to find the f(T) with ε as a parameter that measurs the nature of the interaction between vertices and the surface

• This is the typical f-T curve for ε