Modeling and studying RNA secondary structure Eugne Asarin LIAFA, - - PowerPoint PPT Presentation

Modeling and studying RNA secondary structure

Eugène Asarin LIAFA, CNRS & Univ. Paris Diderot

Credits

Co-authors, partners and teachers:

• Vassily Lyubetsky, Alexander Seliverstov (IITP)
• Thierry Cachat, Tayssir Touili (LIAFA)

Sponsor:

• CNRS/RAS convention d’échanges

EVOLVER/REVERA

Special thanks to:

• Hervé Isambert (I. Curie)

Disclaimer

• I am not a bioinformatician (yet)
• I am (still) a computer scientist with verification

background

– everything is a transition system – one should explore its state space in a smart way

• This talk

– More informatics than byology+physics – More models than solutions – More questions and speculations than answers

Motivating example: Classical Attenuation Regulation

Transcription and Translation

DNA A G C T G C

Transcription and Translation

Polymerase DNA RNA Ribosome Amino Acids Transcription: DNA to RNA, done by Polymerase Translation: RNA to Amino Acids , done by Ribosome Gene

Gene Expression

Gene DNA RNA Ribosome Amino Acids

Gene expressed if Polymerase reach the Gene

Classical Attenuation Regulation

Gene DNA RNA Ribosome

Depends on the structure of the RNA between Ribosome and Polymerase

RNA Secondary Structure

A C A C U G G C U C A C C U U C G G G U G G G C C U U U C U C G

RNA: a sequence of nucleotides A, G, C, U with links A-U and G-C ACACUG C C A C U C G G G U G A G C CUUUCUGCG U U G C Helix

RNA Secondary Structure

(a simplified view)

This structure is dynamic and changes very fast and can cause the slippage of the Polymerase

Polymerase Slippage

Gene RNA Ribosome

T-rich region: connection of Pol and DNA weakens

Polymerase Slippage

Gene RNA Ribosome

T-rich region: connection of Pol and DNA weakens Slippage of the polymerase and the gene is not expressed: Termination

Polymerase Slippage

Gene RNA Ribosome

T-rich region: connection of Pol and DNA weakens Polymerase reaches Gene and the Gene is expressed: Antitermination

Each of these two situations can happen with some probability.

Regulation mechanism : causal chain

• Concentration of a product (say trp)
• Speed of Ribosome
• Dynamics of secondary structure
• Probability of Polymerase slippage
• Gene expression
• (production of trp)

Wanted

• A model of dynamics of the RNA

secondary structure capable to predict the probability of gene expression.

– Should be quantitative – Should represent transient behaviours (steady state not enough) – Should be validated by biological data on regulation

Other motivations

• Other kinds of regulation
• Other alternative behaviors/structures on a

RNA, e.g. in ribozymes

• Scientific curiosity
• Ineresting transfers between Bio and Info

Models, Analyses, Tools

Tools (2 examples)

• Rnamodel - Lyubetsky et al.
• Kinefold – Isambert et al.

The approach: Markov chain

• Features

– The sequence is fixed. – States of the MC : all possible secondary structures on this sequence (or part of it) – Transitions: simple events (see below) – Rates: determined by E

• Difficulties

– As usual, many parameters are difficult to find – The Markov Chain is huge and complex – Only Monte-Carlo simulation is possible – It is still heavy and slow

Some recipes

• Find a succinct structured representation of MC
• Use on-the-fly state generation
• Use symbolic representations
• Use abstractions
• Use acceleration
• Use other advanced trchniques – perfect

simulation etc.

• Think

A succinct representation

We suggest Probabilistic Rewriting Systems

The idea?

• Represent the RNA secondary structure

by a term

• Represent the dynamics of the secondary

structure by rewriting rules

The set of windows

RNA Ribosome Polymerase

w =(R,P) R: position of the Ribosome in the RNA P: position of the Polymerase in the RNA W={w=(R,P) s.t. 13 R P l } l: the length of the regulatory region

The helices

A C B D f = (A,B,C,D)

The hypohelices

A C B D f = (A,B,C,D) g = (E,F,G,H) F E G H

f(g)

The structure as a term

A C B D f = (A,B,C,D) w =(R,P) P R

w(f)

The structure as a term

A C B D f = (A,B,C,D) g = (E,F,G,H) w =(R,P) P R

w(f , g)

E G F H

The structure as a term

f j k h g R P

w =(R,P) w(f , g(h,k) , j) The order is not important

The Dynamics as a Probabilistic Rewriting System

Extension and Reduction of a helix

A C B D H G F E

f = (A,B,C,D) g = (E,F,G,H)

Rewriting rule: f g One rule for multipe contexts

(De)-Composition of a Hypohelix

f j k h

f j k h g

Rewriting rule: w(f , h , k , j) w(f , g(h , k) , j) Meta Rewriting rule: w( , h , k ) w( , g(h , k) )

(De)-Composition of a Helix

f j k h m

(De)-Composition of a Helix

f j k h g m

Meta Rewriting rule: m( , h , k ) m( , g(h , k) )

(De)-Composition of a Helix

One rule for multipe contexts

The Window Movement

RNA Ribosome Polymerase

w =(R,P) Movement of the Ribosome: (R , P)(x) (R+3 , P) (x’) Movement of the Polymerase: (R , P) (x) (R , P+1) (x) Termination: Slippage of the Polymerase: (R , P) (x) three rules for multipe contexts

Rates of the rewriting Rules

• Rates of the Markov Chain determined by
• E. Two terms

– Stacking energy (easy) – Free energy (a bit obscure)

• Movements of Rib and Pol – even more
• bscure

Other optimisations

Implemented or not

On-the-fly (everybody does it)

• States of the MC are created only when

visited.

• Still needed efficient data structures for the

states

• Needed efficient algorithms to find all the

successors of a given state (and the rates)

Concretely

repeat 1000 times s=empty window repeat find all successors s’ of s and rates s->s’ s= a random s’ until expressed or aborted

• utput statistics

Symbolic representation

(not used yet) A data structure (close to a formula) to represent current set of states or probability distribution.

• Very successfull in verification domain
• We tried probabilistic tree automata, they

explode

• Thinking

Abstraction

(everybody use it in a naive way)

• Aim : to have fewer states
• Idea: to group together several states
• Rnamodel and Kinefold : only maximal

helices stored.

• We try to group some close helices:

800000->300000 states

• Abstraction-based algorithms should be

done systematically

Abstraction (what if)

• Biological description is very abstract

(terminator, antiterminator, noise)

• What if … a model is possible at this

level?

• To think

Acceleration

• Problem : many fast transitions without

any progress, major event rare

• Partial solution: group similar states

together

• A better one : Isambert’s clustering
• To think more

Advanced simulation methods

• « Perfect simulation » - nice but only for

steady state

• Other methods from Performance

Evaluation – mostly for N^k

• To read and to think

More complex structures

Other structures

• Helices can be « flipped » -

not a big deal

• They can be pseudoknotted
• Biologically relevant

Consequences for modeling and analysis

• What are legal configurations?
• How to compute free energy?
• Data structure : a set of helices, a tree with

shortcuts, a colored graphs?

• Alphabet and state space much bigger…
• Abstractions more involved.
• Kinefold – yes, Rnamodel – in progress

Dreams

• To find simple abstract models
• To analyze them w/o Monte-Carlo
• To transfer techniques from Verification

and Performance evaluation and backwards

Questions?

Continuous Markov Chain

• Probability for moving from s to s’ within t time

units is:

t s s

e

) ' , (

1

Continuous Markov Chain

• Probability for moving from s to s’ within t time

units is:

• =
• )

' , ( ) ( ), 1 ( ) ( ) ' , (

) (

s s s E e s E s s

t s E