1 The Ribosome Flow Model - PowerPoint Presentation

Slide1

1

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: Slide2

2

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 systemsContinued fractionsSlide3

3

From DNA to Proteins

Transcription

: the cell’s machinery

copies the DNA into mRNAThe mRNA travels from the nucleus to

the cytoplasm

Translation

:

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

Slide4

4

Translation

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

Slide5

5

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

Slide6

6

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 sitesSlide7

Simulating TASEP

7

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. Slide8

8

Analysis of TASEP*

8

*

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

Mean field approximations

Bethe

ansatz

Slide9

Ribosome Flow Model*

*

Reuveni, Meilijson, Kupiec, Ruppin

& Tuller, “Genome-scale analysis of translation elongation with a ribosome flow model”, PLoS

Comput. Biol., 20119A 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. Slide10

Ribosome Flow Model

10

(normalized) number of

ribosomes at site

i

State-variables

:

Parameters:

>0

initiation rate

>0 transition rates between

consecutive sitesSlide11

11

Ribosome Flow ModelSlide12

12

Ribosome Flow Model

Just like TASEP, this encapsulates both

unidirectional movement

and

simple exclusion

. Slide13

Simulation Results

All trajectories emanating from

remain in , and converge to a unique

equilibrium point

e. 13

eSlide14

Analysis of the RFM

Uses tools from:

14

Contraction theory Monotone systems theory Analytic theory of continued fractionsSlide15

Contraction Theory*

The system:

15

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

. Slide16

Contraction Theory

Trajectories contract to each other at

an exponential rate.16

a

b

x(t,0,a)

x(t,0,b)Slide17

Implications of Contraction

1. Trajectories converge to a unique

equilibrium point;17

2. The system

entrains to periodic

excitations. Slide18

Contraction and Entrainment*

Definition

is T-periodic if

18

*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, andSlide19

How to Prove Contraction?

The

Jacobian of is the nxn matrix

19Slide20

How to Prove Contraction?

The infinitesimal distance between

trajectories evolves according to20

This suggests that in order to prove

contraction

we need to (uniformly)

bound

J(x)

. Slide21

How to Prove Contraction?

Let be a

vector norm.21

The induced

matrix norm

is:

The induced

matrix measure

is:Slide22

How to Prove Contraction?

Intuition on the matrix measure:

22

Consider Then to 1

st

order in

soSlide23

Proving Contraction

Theorem

Consider the system23

If

for all then the

Comment 1

: all this works for

system is contracting on K with contraction

rate c.

Comment 2

: is Hurwitz.Slide24

Application to the RFM

For

n=3, 24

a

nd

for the matrix measure induced by

the

L

1

vector norm:

for all

The RFM is on the “

verge of contraction

.”Slide25

RFM is

not Contracting on C

For n=3:

25

so

for

is

singular

and thus

not

Hurwitz. Slide26

Contraction After a

Short Transient (CAST)*

Definition is a CAST if 26

*M., Sontag & Tuller, “

Entrainment to Periodic Initiation and Transition Rates in the Ribosome Flow Model

”,

submitted,

2013.

there exists

such that

-> C

ontraction

after an

arbitrarily small

transient in time and amplitude.

Slide27

Motivation for Contraction after a Short Transient (CAST)

Contraction is used to prove

asymptotic properties (convergence to equilibrium point; entrainment to a periodic

excitation). 27

Slide28

Application to the RFM

Theorem

The RFM is CAST on . 28

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. Slide29

Entrainment in the RFM

29

Slide30

Application to the RFM

Theorem

The RFM is CAST on C. 30

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. Slide31

Entrainment in the RFM

31

Here

n

=3, Slide32

Analysis of the RFM

Uses tools from:

32

Contraction theory Monotone systems theory Analytic theory of continued fractionsSlide33

Monotone Dynamical Systems*

Define a (partial) ordering between vectors

in Rn by:

33

*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.Slide34

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.

34 *Sontag, “Monotone and Near-Monotone Biochemical Networks”, Systems & Synthetic Biology, 2007Slide35

When is a System Monotone?

Theorem

(Kamke Condition.) Suppose that f

satisfies:

35

then is monotone.

Intuition:

assume

monotonicity

is

lost,

thenSlide36

Verifying the

Kamke Condition

Theorem cooperativity Kamke

condition ( system is monotone)

36This means that increasing increases

Definition

is called

cooperative

if Slide37

Application to the RFM

Every off-diagonal entry is non-

negative on C. Thus, the RFM is a cooperative system.

37

Proposition The RFM is monotone on C.

Proof

:

Slide38

RFM is Cooperative

increase. A “traffic jam” in a site induces

“traffic jams” in the neighboring sites. 38

Intuition

if x2 increases then

andSlide39

RFM is Monotone

39

Biological implication: a larger initial

distribution of ribosomes induces a

larger distribution of ribosomes for all time. Slide40

Analysis of the RFM

Uses tools from:

40

Contraction theory Monotone systems theory Analytic theory of continued fractionsSlide41

41

Continued Fractions

Suppose (for simplicity) that

n

=3. Then

Let denote the unique equilibrium point in C. Then Slide42

42

Continued Fractions

This yields:

Every

e

i

can be expressed as a

continued fraction

of

e

3

.

.

.Slide43

43

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

. Slide44

44

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.Slide45

45

HRFM and Periodic Continued Fractions

In the HRFM,

This is a

periodic

continued fraction, and we can say a lot more about

e

. Slide46

46

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. Slide47

mRNA

Circularization*

47

*

Craig,

Haghighat

, Yu &

Sonenberg

, ”Interaction of

Polyadenylate

-Binding Protein with the eIF4G homologue PAIP enhances translation”,

Nature

, 1998 Slide48

RFM as a Control System

This can be modeled by the

RFM with Input and Output (RFMIO):

48

*

Angeli

& Sontag, “Monotone Control Systems”,

IEEE TAC

, 2003

and then closing the loop via

Remark: The RFMIO is a

monotone

control system

.*Slide49

RFM with Feedback*

49

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

.Slide50

RFM with Feedback*

50

Theorem

In the homogeneous case,

where

Biological implication:

may be useful,

perhaps, for re-engineering gene translation. Slide51

Further Research

51

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,….Slide52

Summary

52

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!