A Model of Bacterial Chromosome Architecture Matthew Wright, Daniel Segre, George Church

Ja mie Goodsell

Genomic Scale Structure

Can we understand the 3-d structure of the chromosome? How optimal is the spatial organization of DNA for cell? Can we link function and chromosome structure?

DNA structure has conserved features Hypothesis

Mycoplasma Pneumoniae 816 Kbp 90% Coding 688 Genes 110 Membrane Proteins 52 Ribosomal Proteins No Active Transport No Regulation Limited Metabolism Few DNA Binding Proteins A Model System

.5  m diameter.06  m 3 volume 8000 Ribosomes would fill the cell Extended DNA 80  m in diameter over 100 times cell diameter “Nose” polarity Features

MicroscopyCross-linkingLoop Patterns Tom Knight Gasser et al. Science Dekker etal. Science Empirical Constraints

Transmembrane Proteins Potter MD, Nicchitta CV, 2002 J Biol Chem Jun 28;277(26) 110 genes RNA and or Protein Complexes 52 genes Metabolism DNA Structural Forces Tobias I et al Phys Rev E Stat Phys Plasmas Fluids Relat Intdisc. Topics Jan;61(1) Replication Theoretical Constraints

Symmetry Constraints Symmetric Replication If polymerases replicate at a constant rate symmetric sites from origin are close when replicated Flattened Circle O T

R1R1 R2R2 M1M1 M2M2 M3M3  Cost Function + other terms

Random Walk of GenomeMontecarlo of Parametrized Structures Methods

Random Walk r   n segments 2n-1 Parameters

Montecarlo of Parametrized Structures A Random Walk in Helical Parameter Space

General Helix Parameters a (rise) Supercoil Parameters w (frequency) Ac (amplitude of cos) As (amplitude of sin) Radial Parameters R (maximum large radius) d (frequency of large radial oscillations) Helix Parameters

Energy Decreases

Trivial Solution

Entangled Solution

Possible Solution

Gene Distribution on Structure

Begin With Optimization in Helical Parameter Space Then Perform Random Walk of Genome for Secondary Optimization Generate Relatively Ordered Structures while allowing Local Disorder to Meet Constraints Combine Both Methods

Starting Structure

Final Structure

time steps cost Energy

Prelimary data are promising Incorporate Distance Geometry Need to calculate statistics Gather experimental Data predict and test Incorporate Replication and Dynamics Current

Distance Geometry Represent Structure in terms of distances Constraints fit into a single matrix Matrix with “bounds” defines all possible configurations Can find inconsistencies in constraints Rotationally invariant

Basis Cholesky or eigenvalue decomposition of inner product matrix, M Can get M from D, matrix of distances by defining an origin

Additional Cost Terms Proximity of Enzymes during Metabolism Stoichiometric Matrix Curvature Replication Incorporate Forces on DNA by Using Elastic Rod Model

Classical Model Constraints from Replication Paired Fork Model

Polymerase Based Model Replicate chromosome structure and separate t

If constraints based on function predict structure then structure and function are related at genome scale Potential new class of model Conclusions

Acknowledgements George Church Daniel Segre’ Church Lab

Method Place constraints in matrix Solve for upper and lower bounds from triangle inequalities Randomly choose a configuration within these bounds Embed in 3 dimensions Minimize error

Model for nose replication Seto S, Layh-Schmitt G, Kenri T, Miyata M. J Bacteriol 2001 Mar;183(5): Visualization of the attachment organelle and cytadherence proteins of Mycoplasma pneumoniae by immunofluorescence microscopy.

Bidirectional 2 Polymerase Complexes Remain Attached Daughter DNA Separate Sides Causes Minimal Entanglement Allows for Multiple Firing of Origins Paired fork model

Topological Consequences

Triangle Bound Smoothing Upper bounds Lower bounds

Frenet Frame on Helix

P(i,t) P(i,t+1) P(i-1,t) P(i+1,t) P(i+1,t+1) P(i-1,t+1) dd dd Relaxing the Perturbed Structure

Melting Temperature Short Duplex –C total concentration of single strands Long Duplex

Wordsize (a digression) Blast seeds with at least 7 base string of identities Want to find all alignments with at most 20 mismatches What is the probability of finding a stretch of 7 identities in a string of length 70 with 20 mismatches?

Marbles Maps into the problem of partitioning a string of length 70 into 21 bins Total number of ways etc

Counting Now count the fraction with at least a stretch of 7 But over-counting is a problem

Correcting The cases where 2 bins each have a 7 mer is counted twice so subtract this number once Problem with the cases where there are 3 bins with a 7 mer

Correction Continued Principle of inclusion-exclusion

Extension Coefficients for at least m bins of wordsize l m=2 –1,-2, 3,-4 … m=3 –1,-3,5,-7

A familiar object?

Hello Blaise