1 The Ribosome Flow Model - PowerPoint Presentation

Slide1

1

The Ribosome Flow Model

Michael MargaliotSchool of Electrical Engineering Tel Aviv University, Israel

Tamir

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

Joint work with:

Gilad Poker

Yoram ZaraiSlide2

2

Outline

1. Gene expression and ribosome flow 2. Mathematical models: from TASEP

to the

Ribosome Flow Model (RFM) 3. Analysis of the RFM; biological implicationsSlide3

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, ….3Slide4

Gene Expression: the Central Dogma

Gene (DNA)

Transcription

mRNA

Translation

Protein

4Slide5

Gene Expression

5Slide6

Translation

6

http://www.youtube.com/watch?v=TfYf_rPWUdY Slide7

Translation: the Genetic Code

The mRNA is built of codons

7Slide8

Three Phases of Translation

Initiation:

a ribosome binds to the mRNA strand at a start codon Elongation: tRNA carries the corresponding amino-acid to the ribosome Termination:

ribosome releases amino-acid chain that is then folded into an active protein

8Slide9

Flow of Ribosomes

9

Source:

http

://www.nobelprize.orgSlide10

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.

10* Ingolia, Ghaemmaghami, Newman & Weissman, Science, 2009. * Ingolia, Nature

Reviews

Genetics

,2014.

Slide11

Totally Asymmetric Simple Exclusion Process (TASEP

)*11

A

stochastic

model: particles hop along a lattice of consecutive sites

Movement 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 Slide12

Analysis of TASEP

Rigorous 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.**

12

*Derrida, Evans, Hakim & Pasquier, J. Phys. A: Math., 1993. **Schadschneider, Chowdhury & Nishinari, Stochastic Transport in Complex Systems:

From

Molecules to

Vehicles

,

2010. Slide13

Ribosome Flow Model (RFM)*

Transition

rates: . = initiation rateState variables: , normalized ribosome occupancy level at site i

State space:

13*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

codonsSlide14

14

Ribosome Flow Model

unidirectional

movement

& simple exclusionSlide15

15

Ribosome Flow Model

is

the

translation

rate

at time Slide16

Analysis of the RFM

Based on tools from systems and control

theory: 16

Contraction

theory Monotone systems theory Analytic theory of

continued fractions

Spectral analysis

Convex optimization theory

Random matrix theory

 Slide17

Contraction Theory*

The system:

17is

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.Slide18

Contraction Theory

Trajectories contract to each other at

an exponential rate.18

a

b

x(t,0,a)

x(t,0,b)Slide19

Implications of Contraction

1. Trajectories converge to a unique

equilibrium point (if one exists);19

2. The system

entrains to periodic excitations. Slide20

Contraction and Entrainment*

Definition:

is T-periodic if 20

*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, andSlide21

Proving Contraction

The

Jacobian of is the nxn matrix21Slide22

Proving Contraction

The infinitesimal distance between

trajectories evolves according to22

This suggests that in order to prove

contraction we need to (uniformly)

bound

J(x)

. Slide23

Proving Contraction

Let be a

vector norm.23

The induced

matrix norm is:

The induced

matrix measure

is:Slide24

Proving Contraction

Intuition on the matrix measure:

24

Consider Then to 1

st order in

soSlide25

Proving Contraction

Theorem:

Consider the system25

If

for all then the

Comment 1

: all this works for

system is contracting on K with contraction

rate c.

Comment 2

:

is

Hurwitz.Slide26

Application to the RFM

For

n=3, 26

and for the matrix measure induced by

the

L

1

vector norm: for all

The RFM is on the “

verge of contraction

.”

 Slide27

RFM is

not Contracting on C

For n=3: 27

so for is singular

and thus

not

Hurwitz.

 Slide28

Contraction After a

Short Transient (CAST)*

Definition: is CAST if 28

*Sontag, M., and

Tuller

,

“

On three generalizations of contraction

”,

IEEE CDC

2014.

there exists such that

-> C

ontraction

after an

arbitrarily small

transient in time and amplitude.

Slide29

Motivation for Contraction after a Short Transient (CAST)

Contraction is used to prove

asymptotic properties (convergence to equilibrium point; entrainment to a periodic excitation). 29

Slide30

Application to the RFM

Theorem:

The RFM is CAST on . 30

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. Slide31

Simulation Results

All trajectories emanating

from

C=[0,1]

3remain in C, and converge to a unique equilibrium point e

.

31

eSlide32

Entrainment in the RFM

32

Slide33

Application to the RFM

Theorem:

The RFM is CAST on C. 33

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. Slide34

Entrainment in the RFM

34

Here

n

=3,

Slide35

Analysis of the RFM

Uses tools from:

35

Contraction theory

Monotone systems theory Analytic theory of continued fractions Spectral analysis Convex optimization theory

Random matrix theory,… Slide36

Monotone Dynamical Systems*

Define a (partial) ordering between vectors

in Rn by: .

36

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

Definition

is called

monotone

if

i.e., the dynamics preserves the partial

ordering.Slide37

Monotone Systems

in the Life Sciences*

behavior is ordered and robust with respect to parameter valueslarge systems may be modeled as interconnections of monotone subsystems.37

*

Sontag, “Monotone and near-monotone biochemical networks”, Systems & Synthetic Biology, 2007*Angeli, Ferrell, Sontag, ”Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems”,

PNAS

, 2004

. Slide38

When is a System Monotone?

Theorem:

cooperativity system is monotone 38

This means that increasing increases

Definition:

is called

cooperative

if

.

 Slide39

Application to the RFM

Every off-diagonal entry is non-negative

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

39

Proposition: The RFM is monotone on C.

Proof

:

 Slide40

RFM is Cooperative

increase. A “traffic jam” in a site induces

“traffic jams” in the neighboring sites. 40

Intuition: if

x2 increases then

andSlide41

RFM is Monotone

41

Biological implication: a larger initial

distribution of ribosomes induces a larger distribution of ribosomes for all

time.

x

1

(0)=a

1

x

2

(0)=a

2

,

…

x

1

(0)=b1 x2(0)=b2, …

a≤b x(t,a)≤x(t,b) x1(t,b) x2(t,b),…

x1(t,a) x2(t,a),… Slide42

Analysis of the RFM

42

Contraction theory Monotone systems theory

Analytic theory of

continued fractions Spectral analysis Convex optimization theory Random matrix theory,… Slide43

43

Continued Fractions

Suppose (for simplicity) that

n

=3. Then

Let denote the unique equilibrium point in C. Then Slide44

44

Continued Fractions

This yields:

Every

e

i

can be expressed as a

continued fraction

of

e

3

.Slide45

45

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

.

 Slide46

46

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.Slide47

47

HRFM and Periodic Continued Fractions

In the HRFM,

This is a

1-periodic

continued fraction, and we can say a lot more about

e

3

.

 Slide48

48

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).

 Slide49

mRNA

Circularization*

49

*

Craig, Haghighat, Yu & Sonenberg, ”Interaction of Polyadenylate-Binding Protein with the eIF4G homologue PAIP enhances translation”, Nature, 1998 Slide50

RFM as a Control System

This can be modeled by the

RFM with Input and Output (RFMIO):

50

*

Angeli

& Sontag, “Monotone Control Systems”,

IEEE

TAC

, 2003

and then closing the loop via

Remark: The RFMIO is a

monotone

control system

.*

 Slide51

RFM with Feedback*

51

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.Slide52

HRFM with Feedback

52

Theorem:

In the homogeneous case,

where .

Biological implication:

may be useful,

perhaps, for re-engineering gene translation. Slide53

Analysis of the RFM

Uses tools from:

53

Contraction theory

Monotone systems theory Analytic theory of continued fractions

Spectral analysis

Convex optimization theory

Random matrix theory,… Slide54

54

Recall that

Spectral Analysis

Let

Then

is a solution of

Continued fractions are closely related to

tridiagonal

matrices. This yields a

spectral representation

of the mappingSlide55

55

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 Slide56

Application 1: Concavity

56

Let denote the steady-state translation rate.

Theorem:

is a strictly concave function.

 Slide57

Maximizing Translation Rate

57

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. Slide58

Maximizing Translation Rate*

58

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.Slide59

Maximizing Translation Rate

59Slide60

Application 2: Sensitivity

60

Sensitivity of R to small changes inthe rates -> an eigenvalue sensitivity

problem. Slide61

Application 2: Sensitivity*

61

Theorem

: Suppose that

*

Poker,

M

. and Tuller,

“Sensitivity of mRNA translation,

submitted

,

2014.

Then

=

Rates at the center of the chain are more important. Slide62

Further Research

62

Analysis: controllability andobservability

, 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).Slide63

Conclusions

63

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!