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The Ribosome Flow Model

Michael Margaliot

School of Electrical Engineering

Tel Aviv University, Israel

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

Joint work with:

Gilad Poker Yoram Zarai

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Outline

1. Gene expression and ribosome flow

2. Mathematical models: from TASEP to the

Ribosome Flow Model (RFM)

3. Analysis of the RFM; biological

implications

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Gene Expression

The transformation of the genetic info encoded in the DNA into functioning proteins.

A fundamental biological process: human health, evolution, biotechnology, synthetic biology, ….

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Gene Expression: the Central Dogma

Gene (DNA)

Transcription

mRNA

Translation

Protein

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Gene Expression

Translation

6http://www.youtube.com/watch?v=TfYf_rPWUdY

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Flow of Ribosomes

Source: http://www.nobelprize.org

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The Need for Computational Models of Translation

Expression occurs in all organisms, in almost all cells and conditions. Malfunctions correspond to diseases.

New experimental procedures, like ribosome profiling*, produce more and more data.

Synthetic biology: manipulating the genetic machinery; optimizing translation rate.

* Ingolia, Ghaemmaghami, Newman & Weissman, Science, 2009. * Ingolia, Nature Reviews Genetics ,2014.

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Totally Asymmetric Simple Exclusion Process (TASEP)*

A stochastic model: particles hop along a lattice of consecutive sitesMovement is unidirectional (TA)Particles can only hop to empty sites (SE)*MacDonald & Gibbs, Biopolymers, 1969. Spitzer, Adv. Math., 1970. *Zia, Dong & Schmittmann, “Modeling Translation in Protein Synthesis with TASEP: A Tutorial and Recent Developments”, J Stat Phys , 2010

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Analysis of TASEPRigorous analysis is non trivial. Homogeneous TASEP: steady-state

current and density profiles have been derived using a matrix-product approach.*

TASEP has become a paradigmatic model for non-equilibrium statistical mechanics, used to model numerous natural and artificial processes.***Derrida, Evans, Hakim & Pasquier, J. Phys. A:

Math., 1993. **Schadschneider, Chowdhury & Nishinari, Stochastic Transport in Complex Systems: From Molecules to Vehicles, 2010.

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Ribosome Flow Model (RFM)*

Transition rates: . = initiation rate

State variables: , normalized ribosome occupancy level at site i

State space: *Reuveni, Meilijson, Kupiec, Ruppin & Tuller, “Genome-scale Analysis of Translation Elongation with a Ribosome Flow Model”, PLoS Comput. Biol., 2011

•A deterministic model for ribosome flow •Mean-field approximation of TASEP•mRNA is coarse-grained into n

consecutive sites of codons

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Ribosome Flow Model

unidirectional movement & simple exclusion

1 0 1 1 1 2

2 1 1 2 2 2 3

1 1

1 1

1 1

1

( ) ( )

( ) ( )

( )n n n n n n

x x x x

x x x x x

x x x x

0 1

1x 2x nx

n1n

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Ribosome Flow Model

is the translation rate at time

1 0 1 1 1 2

2 1 1 2 2 2 3

1 1

1 1

1 1

1

( ) ( )

( ) ( )

( )n n n n n n

x x x x

x x x x x

x x x x

( ) : ( )n nR t x t .t

Analysis of the RFMBased on tools from systems and control

theory:

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• Contraction theory• Monotone systems theory • Analytic theory of continued fractions• Spectral analysis• Convex optimization theory• Random matrix theory

⋮

Contraction Theory*

The system:

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is contractive on a convex set K, with

contraction rate c>0, if

for all

*Lohmiller & Slotine, “On Contraction Analysis for Nonlinear

Systems”, Automatica, 1988.*Aminzare & Sontag, “Contraction methods for

nonlinear systems: a brief introduction and some open

problems”, IEEE CDC 2014.

0, , 0.a b K t t

0 0 0| ( , , ) ( , , ) | exp( ( )) | |x t t a x t t b c t t a b

Contraction Theory

Trajectories contract to each other at

an exponential rate.18

a

b

x(t,0,a)

x(t,0,b)

Implications of Contraction

1. Trajectories converge to a unique

equilibrium point (if one exists);

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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

Proving Contraction

The Jacobian of is the nxn matrix

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Proving 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).

Proving Contraction

Theorem: Consider the system

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If for all then the

Comment 1: all this works for

system is contracting on K with contraction

rate c.

Comment 2: is Hurwitz.0( ( )) ( )J x J x

Application to the RFM

For 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.”

J ( x )=(− 𝜆0− 𝜆1(1−𝑥1) 𝜆1𝑥1 0𝜆1(1−𝑥1) − 𝜆1𝑥1− 𝜆2(1− 𝑥3) 𝜆2𝑥2

0 𝜆2(1−𝑥3) − 𝜆2 𝑥2− 𝜆3)

RFM is not Contracting on C

For n=3:

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so for is singular

and thus not Hurwitz.

J ( x )=(− 𝜆0− 𝜆1(1−𝑥1) 𝜆1𝑥1 0𝜆1(1−𝑥1) − 𝜆1𝑥1− 𝜆2(1− 𝑥3) 𝜆2𝑥2

0 𝜆2(1−𝑥3) − 𝜆2 𝑥2− 𝜆3)

Contraction After a Short Transient (CAST)*

Definition: is CAST if

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*Sontag, M., and Tuller, “On three generalizations of contraction”, IEEE CDC 2014 .

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. and 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.

Simulation Results

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

0(0) .x x

All trajectories emanating from C=[0,1]3

remain in C, and converge to a unique

equilibrium point e. 31

0.ft e

Entrainment in the RFM

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0

Application to the RFMTheorem: The RFM is CAST on C.

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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, and Tuller, “Entrainment to Periodic Initiation and Transition Rates in the Ribosome Flow Model”, PLOS ONE, 2014 .

Entrainment in the RFM

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Here n=3,

0 2 2( ) sin( ),t t 1 1( ) ,t

2

13 2

2( ) sin( ),t t 3

14 2 2

8( ) cos( ).t t

Analysis of the RFM

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• Contraction theory• Monotone systems theory • Analytic theory of continued fractions• Spectral analysis• Convex optimization theory• Random matrix theory,…

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

Let denote the unique equilibrium point in C. Then

1 0 1 1 1 2

2 1 1 2 2 2 3

3 2 2 3 3 3

1 1

1 1

1

( ) ( )

( ) ( )

( ) .

x x x x

x x x x x

x x x x

0 1 1 1 2

2 2 3

3 3

1 1

1

( ) ( )

( )

.

e e e

e e

e

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Continued Fractions

This yields:

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

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Continued Fractions

Furthermore, e3 satisfies:

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

𝜆3𝜆0𝑒3=1−

𝜆3𝑒3

𝜆1(1− 𝜆3𝑒3𝜆2(1−𝑒3))

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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 1-periodic continued fraction, and we can say a lot more about e3.

𝜆𝑐

𝜆0𝑒3=1−

𝑒3

1−𝑒31−𝑒3

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Equilibrium Point in the HRFM*

Theorem: In the HRFM,

*M. and 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 (assuming homogeneous transition rates).

lim𝜆0→∞

𝑒𝑛=1

4𝑐𝑜𝑠2( 𝜋𝑛+2 )❑

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.*

𝜆0→𝑢 (𝑡 ) ,

RFM with Feedback*

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Theorem: The closed-loop system admits

an equilibrium point e that is globally

attracting in C.

*M. and Tuller, “Ribosome Flow Model with Feedback”, J. Royal Society Interface, 2013

Biological implication: as before, but this

is probably a better model for translation

in eukaryotes.

Analysis of the RFM

Uses tools from:

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• Contraction theory• Monotone systems theory • Analytic theory of continued fractions• Spectral analysis• Convex optimization theory• Random matrix theory,…

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Recall that Spectral Analysis

𝑅 (𝑡 )=𝜆𝑛𝑥𝑛 (𝑡 ) . Let

Then 𝑅¿𝑅 (𝜆0 ,𝜆1 ,…,𝜆¿¿𝑛)¿ is a solution of

0=1−𝑅/ 𝜆0

1−𝑅 /𝜆1

1−𝑅/ 𝜆2⋱ ❑1−𝑅/𝜆𝑛

Continued fractions are closely related to tridiagonal matrices. This yields a spectral representation of the mapping

0 1( , , ..., ) .n R

: .n nR e

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Theorem: Consider the (n+2)x(n+2) symmetric, non-negative and irreducible tridiagonal matrix:

Spectral Analysis*

Denote its eigenvalues by . Then

A spectral representation of

0 1( , , ..., ) .n R

1 22

/ .n R

Application 1: Concavity

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Let denote the steady-state

translation rate.

: n nR e

Theorem: is a strictly concave function.

𝑅¿𝑅 (𝜆0 ,𝜆1 ,…,𝜆¿¿𝑛)¿

Maximizing Translation Rate

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Translation is one of the most energy consuming processes in the cell. Evolution optimized this process, subject to the limited biocellular budget.

Maximizing translation rate is also important in biotechnology.

Maximizing Translation Rate*

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0 1Max ( , ,..., )

Sub : 0n

i

R

0 0 1 1 ... n nw w w b

Since R is a concave function, this is a convex optimization problem.

- A unique optimal solution- Efficient algorithms that scale well with n

Poker, Zarai, M. and Tuller,”Maximizing protein translation rate in the non-homogeneous ribosome flow model: a convex optimization approach”, J. Royal Society Interface, 2014.

Maximizing Translation Rate

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* *0 0 1 1 .w w b

Application 2: Sensitivity

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Sensitivity of R to small changes in

the rates -> an eigenvalue sensitivity

problem.

Application 2: Sensitivity*

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Theorem: Suppose that

*Poker, M. and Tuller, “Sensitivity of mRNA translation, submitted, 2014.

Then

=log (𝜕𝑅 /𝜕 𝜆𝑖)

Rates at the center of the chain are more

important.

3

1 2

3 3

2 33

sin( )sin( ).

( )cos ( )i

i iR n n

nn

Further Research

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1. Analysis: controllability and

observability, stochastic rates, networks

of RFMs,…

3. TASEP has been used to model:

biological motors, surface growth, traffic

flow, ants moving along a trail, Wi-Fi

networks,….

2. Modifying the RFM (extended objects,

ribosome drop-off).

Conclusions

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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!