Epigenetic Modifications as dilute one dimensional Spin Glass systems

Ritam Das (MS19029)

IDC451

Epigenetic Modifications as dilute one dimensional Spin Glass systems

Epigenetic Modifications as dilute one dimensional Spin Glass systems

Epigenetic Modifications as dilute one dimensional Spin Glass systems

Epigenetic Modifications as dilute one dimensional Spin Glass systems

  • Magnetic systems characterised by disorder and frustration

  • Disorder in spins is analogous to positional disorder in chemical glass hence the name 'glass'

Disorder in Spin Glass Systems

Ferromagnetic

Interactions

Antiferromagnetic

Interactions

Disorder in Spin Glass Systems

Ferromagnetic

Interactions

Antiferromagnetic

Interactions

Disorder

F              F             F              F              F

AF           AF          AF            AF          AF

AF         AF          AF            AF          AF

AF           AF          AF           AF            AF

F              F             F              F              F

F              F             F              F              F

Frustration in Spin Glass Systems

Unfrustrated

Spins

Geometric

Frustration

Spin Glass

Frustration

AF

AF

AF

AF

AF

AF

AF

AF

AF

F

AF

Quantifying the Interactions

\mathcal{H} = -J_{ij} \sigma_i\sigma_j
J_{ij} > 0 \text{ : Ferromagnetic}
J_{ij} < 0 \text{ : Antiferromagnetic}
\text{Interaction Matrix}

}

}

\text{Spin States}

Quantifying the Interactions

\mathcal{H} = -J_{ij} \sigma_i\sigma_j
J_{ij} > 0 \text{ : Ferromagnetic}
J_{ij} < 0 \text{ : Antiferromagnetic}

Multivalley Potential Landscape

Power of Spin Glasses

Depending on the choice of the J, spin glasses can be used to model a number of systems.

What are Epigenetic Modifications and why do we care?

Epigenetic Modifications 

Today, this term is used to refer to heritable alterations that are not due to changes in DNA sequence

Epigenetic Modifications (historically) refer to the complex interactions between the genome and the environment

DNA

Methyl Group

Chromatin

Chromosome

Histone Tails

Histone

Epigenetic Factor

Folding your Genes

Epigenetic Modifications Activate or Silence Genes

Why do we care?

Loci are not independent of each other

Modifications at one site influence other loci in proximity

Proximity in Cartesian Space

Proximity in Genetic Space

\neq

The Problem

How do we quantify this influence of one site over the others?

Genes as Spins

*Inaccuracy for the sake of rhyme

*

Genes as Spins

*Inaccuracy for the sake of rhyme

*

States of epigenetic modifications are our spin states

Probability of contact between loci determines interaction matrix J

(\sigma)

The Spin States

\sigma
+1
-1
0

Type 1 Modification

Type 2 Modification

No Modification

One Dimensional Lattice

1
-1
0
1
1
-1
...
0
-1
1
L

The Interaction Matrix

Contact Probability Distribution

Fission Yeast

Budding Yeast

Human

(From HiC Data)

The Interaction Matrix

Probability of contact -

p(r)

- distance between two loci

r
J_{ij} =
\{
\pm 1, \text{with probability } p(r)
0\text{, otherwise}

The Hamiltonian

\sigma_i = \{+1, 0,-1\}
\mathcal{H} = -J_{ij} \sigma_i\sigma_j

  • Navigate through the rugged energy landscape

  • Find a (relatively) stable configuration

Our Goal

Simulations

  • Monte Carlo Simulations
  • Parallel Tempering Algorithm
  • Two identical copies simulated parallely

Monte Carlo Algorithm

Start Random

Flip A Spin

ΔE < 0

Accept the change

Reject the change

Y

N

p > \min{(1, e^{-\beta\Delta H})}

N

Y

Monte Carlo Simulations using parallel tempering gives us an idea about the potential landscape for the system

Monte Carlo Simulations using parallel tempering gives us an idea about the potential landscape for the system

Monte Carlo Simulations using parallel tempering gives us an idea about the potential landscape for the system

Avenues to explore

  • Appearance of glassy phase
  • Ageing in spin glass vs Biological Ageing
  • Addition of chemical potential to the Hamiltonian
\mathcal{H} = -J_{ij} \sigma_i\sigma_j + \Delta \sigma_i^2

References

  • Koji Hukushima and Koji Nemoto. Exchange monte carlo method and application to spin glass simulations. Journal of the Physical Society of Japan, 65(6):1604–1608, 1996.

  • G Kotliar, PW Anderson, and Daniel L Stein. One-dimensional spin-glass model with long-range random interactions. Physical Review B, 27(1):602, 1983.

  • L. Leuzzi, G. Parisi, F. Ricci-Tersenghi, and J. J. Ruiz-Lorenzo. Dilute one-dimensional spin glasses with power law decaying interactions. Phys. Rev. Lett., 101:107203, Sep 2008.

  • Fabrizio Olmeda, Tim Lohoff, Stephen Clark, Laura Benson, Felix Krueger, Wolf Reik, and Steffen Rulands. Inference of emergent spatio-temporal processes in single-cell genomics. In APS March Meeting Abstracts, volume 2021, pages S13–009, 2021.

  • Guang Shi and D. Thirumalai. From hi-c contact map to three-dimensional organization of interphase human chromosomes. Phys. Rev. X, 11:011051, Mar 2021.

Thank You

Questions?

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