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The Ribosome Flow Model

Michael MargaliotSchool of Elec. Eng. Tel Aviv University, Israel

Tamir Tuller (Tel Aviv University) Eduardo D. Sontag (Rutgers University)

Joint work with:

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Overview Ribosome flow Mathematical models: from TASEP to the Ribosome Flow Model (RFM)

Analysis of the RFM+biological implications: Contraction (after a short time) Monotone systems Continued fractions

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From DNA to ProteinsTranscription: the cell’s machinery copies the DNA into mRNA

The mRNA travels from the nucleus tothe cytoplasm

Translation: ribosomes “read” the mRNA and produce a corresponding chain of amino-acids

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Translation

http://www.youtube.com/watch?v=TfYf_rPWUdY

http://www.youtube.com/watch?v=TfYf_rPWUdY

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Ribosome FlowDuring translation several ribosomesread the same mRNA. Ribosomes follow each other like cars traveling along a road.

Mathematical models for ribosome flow: TASEP* and the RFM.

*Zia, Dong, Schmittmann, “Modeling Translation in Protein Synthesis with TASEP: A Tutorial and Recent Developments”, J. Statistical Physics, 2011

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Totally Asymmetric Simple Exclusion Process (TASEP)

Particles can only hop to empty sites (SE)Movement is unidirectional (TA)

A stochastic model: particles hop along a lattice of consecutive sites

Simulating TASEP

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At each time step, all the particles are scanned and hop with probability , if the consecutive site is empty.

This is continued until steady state.

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Analysis of TASEP*

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*Schadschneider, Chowdhury & Nishinari, Stochastic Transport in Complex Systems: From Molecules to Vehicles, 2010.

1. Mean field approximations

2. Bethe ansatz

Ribosome Flow Model*

*Reuveni, Meilijson, Kupiec, Ruppin & Tuller, “Genome-scale analysis of translation elongation with a ribosome flow model”, PLoS Comput. Biol., 2011 9

A deterministic model for ribosome flow.

mRNA is coarse-grained into consecutive sites.

Ribosomes reach site 1 with rate , but can only bind if the site is empty.

Ribosome Flow Model

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(normalized) number of ribosomes at site i

State-variables:

Parameters: >0 initiation rate

>0 transition rates between consecutive sites

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Ribosome Flow Model

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Ribosome Flow Model

Just like TASEP, this encapsulates both unidirectional movement and simple exclusion.

Simulation Results

( ) | ( ; ) | .fJ u x t u

0(0) .x x

All trajectories emanating fromremain in , and converge to a unique equilibrium point e. 13

0.ft e

Analysis of the RFM

Uses tools from:

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Contraction theory Monotone systems theory Analytic theory of continued fractions

Contraction Theory*The system:

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is contracting on a convex set K, with contraction rate c>0, if

for all

*Lohmiller & Slotine, “On Contraction Analysis

for Nonlinear Systems”, Automatica, 1988.

Contraction Theory

Trajectories contract to each other at an exponential rate.

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a

b

x(t,0,a)

x(t,0,b)

Implications of Contraction

1. Trajectories converge to a unique equilibrium point;

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2. The system entrains to periodic excitations.

Contraction and Entrainment*Definition is T-periodic if

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*Russo, di Bernardo, Sontag, “Global Entrainment of Transcriptional Systems to Periodic Inputs”, PLoS Comput. Biol., 2010 .

Theorem The contracting and T-periodic system admits a unique periodic solution of period T,

and

How to Prove Contraction?

The Jacobian of is the nxn matrix

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How to Prove Contraction?

The infinitesimal distance between trajectories evolves according to

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This suggests that in order to prove contraction we need to (uniformly) bound J(x).

How to Prove Contraction?Let be a vector norm.

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The induced matrix norm is:

The induced matrix measure is:

How to Prove Contraction?Intuition on the matrix measure:

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Consider Then to 1st order in

so

Proving ContractionTheorem Consider the system

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If for all then the

Comment 1: all this works for

system is contracting on K with contractionrate c.

Comment 2: is Hurwitz.

Application to the RFMFor n=3,

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and for the matrix measure induced by the L1 vector norm: for all

The RFM is on the “verge of contraction.”

RFM is not Contracting on CFor n=3:

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so for is singular

and thus not Hurwitz.

Contraction After a Short Transient (CAST)*

Definition is a CAST if

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*M., Sontag & Tuller, “Entrainment to Periodic Initiation and Transition Rates in the Ribosome Flow Model”, submitted, 2013 .

there exists such that

-> Contraction after an arbitrarily small transient in time and amplitude.

Motivation for Contraction after a Short Transient (CAST)

Contraction is used to prove asymptotic properties (convergence to equilibrium point; entrainment to a periodic excitation).

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Application to the RFMTheorem The RFM is CAST on

.

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Corollary 1 All trajectories converge to a unique equilibrium point e.*

*M.& Tuller, “Stability Analysis of the Ribosome Flow Model”, IEEE TCBB, 2012 .

Biological interpretation: the parameters determine a unique steady-state of ribosome distributions and synthesis rate; not affected by perturbations.

Entrainment in the RFM

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Application to the RFMTheorem The RFM is CAST on C.

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Corollary 2 Trajectories entrain toperiodic initiation and/or transition rates (with a common period T).* Biological interpretation: ribosome distributions and synthesis rate converge to a periodic pattern, with period T.

*M., Sontag & Tuller, “Entrainment to Periodic Initiation and Transition Rates in the Ribosome Flow Model”, submitted, 2013 .

Entrainment in the RFM

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Here n=3,

Analysis of the RFM

Uses tools from:

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Contraction theory Monotone systems theory Analytic theory of continued fractions

Monotone Dynamical Systems*Define a (partial) ordering between vectors in Rn by:

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*Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, AMS, 1995

Definition is called monotone if

i.e., the dynamics preserves the partial ordering.

Monotone Dynamical Systems in the Life Sciences

Used for modeling a variety of biochemical networks:* - behavior is ordered and robust with

respect to parameter values- large systems may be modeled as

interconnections of monotone subsystems.

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*Sontag, “Monotone and Near-Monotone Biochemical Networks”, Systems & Synthetic Biology, 2007

When is a System Monotone? Theorem (Kamke Condition.) Suppose that f satisfies:

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then is monotone.

Intuition: assume monotonicity is lost, then

Verifying the Kamke Condition

Theorem cooperativity Kamke condition ( system is monotone)

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This means that increasing increases

Definition is called cooperative if

Application to the RFM

Every off-diagonal entry is non-negative on C. Thus, the RFM is a cooperative system.

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Proposition The RFM is monotone on C. Proof:

RFM is Cooperative

increase. A “traffic jam” in a site induces “traffic jams” in the neighboring sites.

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Intuition if x2 increases then

and

RFM is Monotone

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Biological implication: a larger initial distribution of ribosomes induces a larger distribution of ribosomes for all time.

Analysis of the RFM

Uses tools from:

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Contraction theory Monotone systems theory Analytic theory of continued fractions

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Continued FractionsSuppose (for simplicity) that n =3. Then

Let denote the unique equilibrium point in C. Then

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Continued Fractions

This yields:

Every ei can be expressed as a continued fraction of e3 .

..

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Continued FractionsFurthermore, e3 satisfies:

....

This is a second-order polynomial equation in e3. In general, this is a th–order polynomial equation in en.

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Homogeneous RFM In certain cases, all the transition rates are approximately equal.* In the RFM this can be modeled by assuming that

*Ingolia, Lareau & Weissman, “Ribosome Profiling of Mouse Embryonic Stem Cells Reveals the Complexity and Dynamics of Mammalian Proteomes”, Cell, 2011

This yields the Homogeneous Ribosome Flow Model (HRFM). Analysis is simplified because there are only two parameters.

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HRFM and Periodic Continued Fractions

In the HRFM,

This is a periodic continued fraction, and we can say a lot more about e.

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Equilibrium Point in the HRFM*Theorem In the HRFM,

*M. & Tuller, “On the Steady-State Distribution in the Homogeneous Ribosome Flow Model”, IEEE TCBB, 2012

Biological interpretation: This provides an explicit expression for the capacity of a gene.

mRNA Circularization*

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*Craig, Haghighat, Yu & Sonenberg, ”Interaction of Polyadenylate-Binding Protein with the eIF4G homologue PAIP enhances translation”, Nature, 1998

RFM as a Control SystemThis can be modeled by the RFM with Input and Output (RFMIO):

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*Angeli & Sontag, “Monotone Control Systems”, IEEE TAC, 2003

and then closing the loop via

Remark: The RFMIO is a monotone control system.*

RFM with Feedback*

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Theorem The closed-loop system admits an equilibrium point e that is globally attracting in C.

*M. & Tuller, “Ribosome Flow Model with Feedback”, J. Royal Society -Interface, to appear

Biological implication: as before, but this is probably a better model for translation in eukaryotes.

RFM with Feedback*

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Theorem In the homogeneous case,

where

Biological implication: may be useful, perhaps, for re-engineering gene

translation.

Further Research

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1. Analyzing translation: sensitivity analysis; optimizing translation rate; adding features (e.g. drop-off); estimating initiation rate;…

2. TASEP has been used to model: biological motors, surface growth, traffic flow, walking ants, Wi-Fi networks,….

Summary

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The Ribosome Flow Model is: (1) useful; (2) amenable to analysis. Papers available on-line at: www.eng.tau.ac.il/~michaelm

Recently developed techniques provide more and more data on the translation process. Computational models are thus becoming more and more important.

THANK YOU!