The Ribosome Flow Model Michael Margaliot School of Elec. Eng. Tel Aviv University, Israel Joint work with: Tamir Tuller (Tel Aviv University) Eduardo D. Sontag (Rutgers University)

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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 Proteins Transcription : the cell’s machinery copies the DNA into mRNA The mRNA travels from the nucleus to the 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 4

Ribosome Flow During translation several ribosomes read 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 5

Totally Asymmetric Simple Exclusion Process (TASEP) A stochastic model: particles hop along a lattice of consecutive sites Movement is unidirectional (TA) Particles can only hop to empty sites (SE)

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Simulating TASEP 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* 1. Mean field approximations 2. Bethe ansatz *Schadschneider, Chowdhury & Nishinari,

Stochastic Transport in Complex Systems: From

Molecules to Vehicles, 2010.

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Ribosome Flow Model*

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.

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

Ribosome Flow Model State-variables

:

(normalized) number of ribosomes at site i

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 .

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Simulation Results

x

(0) 

x

0 .

t f

 0.



e

x t u f

All trajectories emanating from remain in , and converge to a unique equilibrium point e.

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Analysis of the RFM Uses tools from: Contraction theory Monotone systems theory Analytic theory of continued fractions

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Contraction Theory* The system: 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 . 15

Contraction Theory a x(t,0,a) b x(t,0,b) Trajectories contract to each other at an exponential rate.

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Implications of Contraction 1. Trajectories converge to a unique equilibrium point; 2. The system entrains to periodic excitations.

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Contraction and Entrainment* Definition is T-periodic if Theorem The contracting system and T-periodic

admits a unique periodic solution of period T, and *Russo, di Bernardo, Sontag, “Global Entrainment of Transcriptional Systems to Periodic Inputs”, PLoS Comput. Biol., 2010. 18

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

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How to Prove Contraction?

Let be a vector norm .

The induced matrix norm is: The induced matrix measure is:

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How to Prove Contraction?

Intuition on the matrix measure: Consider Then to 1 st order in so

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Proving Contraction Theorem Consider the system If for all then the system is contracting on K with contraction rate c.

Comment 1 : all this works for Comment 2 : is Hurwitz.

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Application to the RFM For n=3, and for the matrix measure induced by the L

1

vector norm: for all The RFM is on the “ verge of contraction .

”

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RFM is not Contracting on C For n=3: so for and thus not Hurwitz. is singular

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Contraction After a Short Transient (CAST)* Definition is a CAST if there exists such that -> Contraction after an arbitrarily small transient in time and amplitude.

*M., Sontag & Tuller, “Entrainment to Periodic Initiation and Transition Rates in the Ribosome Flow Model”, submitted, 2013. 26

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 RFM Theorem The RFM is CAST on . All trajectories converge to a unique equilibrium point e.* Biological interpretation: the parameters determine a unique steady-state of ribosome distributions and synthesis rate; not affected by perturbations.

*M.& Tuller, “Stability Analysis of the Ribosome Flow Model”, IEEE TCBB, 2012. 28

Entrainment in the RFM

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Application to the RFM Theorem The RFM is CAST on C. Corollary 2 Trajectories entrain to periodic 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. 30

Entrainment in the RFM Here n=3,

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Analysis of the RFM Uses tools from: Contraction theory Monotone systems theory Analytic theory of continued fractions

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Monotone Dynamical Systems* Define a (partial) ordering between vectors in R

n

by: Definition is called monotone if i.e., the dynamics preserves the partial ordering.

*Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems AMS, 1995 33 ,

Monotone Dynamical Systems in the Life Sciences Used for modeling a variety of

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biochemical networks:* behavior is ordered and robust with respect to parameter values

-

large systems may be modeled as interconnections of monotone subsystems.

*Sontag, “Monotone and Near-Monotone Biochemical Networks”, Systems & Synthetic Biology , 2007 34

When is a System Monotone? Theorem (Kamke Condition.) Suppose that f satisfies: then is monotone. Intuition: assume monotonicity is lost, then

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Verifying the Kamke Condition Definition if is called cooperative This means that increasing increases Theorem

cooperativity

Kamke condition ( system is monotone)

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Application to the RFM Proposition The RFM is monotone on C. Proof

:

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

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RFM is Cooperative Intuition if x

2

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

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

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Analysis of the RFM Uses tools from: Contraction theory Monotone systems theory Analytic theory of continued fractions

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Continued Fractions Suppose (for simplicity) that n =3. Then Let denote the unique equilibrium point in C. Then

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.

.

Continued Fractions This yields: Every e

i

can be expressed as a continued fraction of e

3

.

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...

.

Continued Fractions Furthermore, e

3

satisfies: This is a second-order polynomial equation in e

3

.

In general, this is a th –order polynomial equation in e

n

.

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

*Ingolia, Lareau & Weissman, “Ribosome Profiling of Mouse Embryonic Stem Cells Reveals the Complexity and Dynamics of Mammalian Proteomes”, Cell , 2011 44

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

*M. & Tuller, “On the Steady-State Distribution in the Homogeneous Ribosome Flow Model”, 2012 IEEE TCBB , 46

mRNA Circularization *

* Craig, Haghighat, Yu & Sonenberg, ”Interaction of Polyadenylate-Binding Protein with the eIF4G homologue PAIP enhances translation”, Nature , 1998 47

RFM as a Control System This can be modeled by the RFM with Input and Output ( RFMIO ): and then closing the loop via Remark: The RFMIO is a monotone control system .*

*Angeli & Sontag, “Monotone Control Systems”, IEEE TAC , 2003 48

RFM with Feedback* Theorem The closed-loop system admits an equilibrium point e that is globally attracting in C. Biological implication: as before, but this is probably a better model for translation in eukaryotes

.

*M. & Tuller, “Ribosome Flow Model with Feedback”, J. Royal Society -Interface, to appear 49

RFM with Feedback* Theorem In the homogeneous case, where Biological implication: may be useful,

Further Research 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, ….

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

The Ribosome Flow Model is: (1) useful; (2) amenable to analysis. Papers available on-line at:

www.eng.tau.ac.il/~michaelm

THANK YOU!

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