The Ribosome Flow Model
Michael Margaliot
School of Electrical Engineering
Tel Aviv University, Israel
Joint work with:
Gilad Poker
Yoram Zarai
Tamir Tuller (Tel Aviv University)
Eduardo D. Sontag (Rutgers University)
1
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
2
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, ….
3
Gene Expression: the Central Dogma
Transcription
Translation
mRNA
Protein
Gene (DNA)
4
Gene Expression
5
Translation
http://www.youtube.com/watch?v=TfYf_rPWUdY
6
Flow of Ribosomes
Source:
http://www.nobelprize.org
9
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.
10
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)
*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
11
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.**
*Derrida, Evans, Hakim & Pasquier, J. Phys. A: Math., 1993.
**Schadschneider, Chowdhury & Nishinari, Stochastic
Transport in Complex Systems: From Molecules to Vehicles,
2010.
12
Ribosome Flow Model (RFM)*
• A deterministic model for ribosome flow
• Mean-field approximation of TASEP
• mRNA is coarse-grained into n consecutive



sites of codons
Transition rates:
.
= initiation rate
State variables:
, normalized
ribosome occupancy level at site i
State space:
*Reuveni, Meilijson, Kupiec, Ruppin & Tuller, “Genomescale Analysis of Translation Elongation with a Ribosome
Flow Model”, PLoS Comput. Biol., 2011
13
Ribosome Flow Model
x 1   0 (1  x 1 )   1 x 1 (1  x 2 )
x 2   1 x 1 (1  x 2 )   2 x 2 (1  x 3 )
x n   n  1 x n  1 (1  x n )   n x n
0
 n 1
1
n
Site 3
x1
x2
Codon
xn
unidirectional movement & simple exclusion14
Ribosome Flow Model
x 1   0 (1  x 1 )   1 x 1 (1  x 2 )
x 2   1 x 1 (1  x 2 )   2 x 2 (1  x 3 )
x n   n  1 x n  1 (1  x n )   n x n
R ( t ) :  n x n ( t )
is the translation rate at time t .
15
Analysis of the RFM
Based on tools from systems and control
theory:
•
•
•
•
•
•
Contraction theory
Monotone systems theory
Analytic theory of continued fractions
Spectral analysis
Convex optimization theory
Random matrix theory
⋮
16
Contraction Theory*
The system:
is contractive on a convex set K, with
contraction rate c>0, if
| x ( t , t 0 , a )  x ( t , t 0 , b ) | exp(  c ( t  t 0 )) | a  b |
for all a , b  K , t  t 0  0.
*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.
17
Contraction Theory
a
x(t,0,a)
b
x(t,0,b)
Trajectories contract to each other at
an exponential rate.
18
Implications of Contraction
1. Trajectories converge to a unique
equilibrium point (if one exists);
2. The system entrains to periodic
excitations.
19
Contraction and Entrainment*
Definition:
is T-periodic if
Theorem : The contracting and T-periodic
system
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.
20
Proving Contraction
The Jacobian of
is the nxn matrix
21
Proving Contraction
The infinitesimal distance between
trajectories evolves according to
This suggests that in order to prove
contraction we need to (uniformly)
bound J(x).
22
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:  ( J ( x ))  0  J ( x ) is Hurwitz.
25
Application to the RFM
For n=3,
− 𝜆0 − 𝜆1 (1 − 𝑥1 )
𝜆1 (1 − 𝑥1 )
J x =
0
𝜆1 𝑥1
− 𝜆1 𝑥1 − 𝜆2 (1 − 𝑥3 )
𝜆2 (1 − 𝑥3 )
0
𝜆2 𝑥2
− 𝜆2 𝑥2 − 𝜆3
and for the matrix measure induced by
the L1 vector norm:
for all
The RFM is on the “verge of contraction.”
26
RFM is not Contracting on C
For n=3:
− 𝜆0 − 𝜆1 (1 − 𝑥1 )
𝜆1 (1 − 𝑥1 )
J x =
0
𝜆1 𝑥1
− 𝜆1 𝑥1 − 𝜆2 (1 − 𝑥3 )
𝜆2 (1 − 𝑥3 )
so for
0
𝜆2 𝑥2
− 𝜆2 𝑥2 − 𝜆3
is singular
and thus not Hurwitz.
27
Contraction After a
Short Transient (CAST)*
Definition:
is CAST if
there exists
such that
-> Contraction after an arbitrarily small
transient in time and amplitude.
*Sontag, M., and Tuller, “On three generalizations
of contraction”, IEEE CDC 2014.
28
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
Application to the RFM
Theorem: The RFM is CAST on
.
Corollary 1: All trajectories converge to a
unique equilibrium point e.*
Biological interpretation: the parameters
determine a unique steady-state of
ribosome distributions and synthesis
rate.
*M. and Tuller, “Stability Analysis of the Ribosome
Flow Model”, IEEE TCBB, 2012.
30
Simulation Results
x (0)  x 0 .
t f  0.
J ( u ) e
| x (t f ; u ) | .
All trajectories emanating from C=[0,1]3
remain in C, and converge to a unique
equilibrium point e.
31
Entrainment in the RFM
0
32
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, and Tuller, “Entrainment to Periodic
Initiation and Transition Rates in the Ribosome
Flow Model”, PLOS ONE, 2014.
33
Entrainment in the RFM
Here n=3,  0 ( t )  2  sin ( 2  t ),  1 ( t )  1,
 2 ( t )  3  sin ( 2  t 
1
2
),  3 ( t )  4  2 cos( 2  t 
1
8
).
34
Analysis of the RFM
•
•
•
•
•
•
Contraction theory
Monotone systems theory
Analytic theory of continued fractions
Spectral analysis
Convex optimization theory
Random matrix theory,…
42
Continued Fractions
Suppose (for simplicity) that n =3. Then
x 1   0 (1  x 1 )   1 x 1 (1  x 2 )
x 2   1 x 1 (1  x 2 )   2 x 2 (1  x 3 )
x 3   2 x 2 (1  x 3 )   3 x 3 .
Let
denote the unique equilibrium
point in C. Then  0 (1  e1 )   1e1 (1  e 2 )
  2 e 2 (1  e 3 )
  3e3 .
43
Continued Fractions
This yields:
Every ei can be expressed as a
continued fraction of e3 .
44
Continued Fractions
Furthermore, e3 satisfies:
𝜆3
𝑒3 = 1 −
𝜆0
𝜆
1
𝜆3 𝑒3
𝜆3 𝑒3
1−
𝜆2 (1 − 𝑒3 )
This is a second-order polynomial
equation in e3.
In general, this is a
polynomial equation in en.
th–order
45
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
46
HRFM and Periodic
Continued Fractions
In the HRFM,
𝜆𝑐
𝑒3
𝑒3 = 1 −
𝑒3
𝜆0
1−
1 − 𝑒3
This is a 1-periodic continued fraction,
and we can say a lot more about e3.
47
Equilibrium Point in the HRFM*
Theorem: In the HRFM,
lim 𝑒𝑛 =
1
𝜋
𝑛+2
Biological interpretation: This provides
an explicit expression for the capacity
of a gene (assuming homogeneous
transition rates).
𝜆0 →∞
4𝑐𝑜𝑠 2
*M. and Tuller, “On the Steady-State Distribution in the
Homogeneous Ribosome Flow Model”, IEEE TCBB, 2012
48
mRNA Circularization*
*Craig, Haghighat, Yu & Sonenberg, ”Interaction of
Polyadenylate-Binding Protein with the eIF4G homologue PAIP
49
enhances translation”, Nature, 1998
RFM as a Control System
This can be modeled by the RFM with
Input and Output (RFMIO):
𝜆0 → 𝑢 𝑡 ,
and then closing the loop via
Remark: The RFMIO is a monotone
control system.*
*Angeli & Sontag, “Monotone Control Systems”,
IEEE TAC, 2003
50
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. and Tuller, “Ribosome Flow Model with
Feedback”, J. Royal Society Interface, 2013
51
Analysis of the RFM
Uses tools from:
•
•
•
•
•
•
Contraction theory
Monotone systems theory
Analytic theory of continued fractions
Spectral analysis
Convex optimization theory
Random matrix theory,…
53
Spectral Analysis
Recall that 𝑅 𝑡 = 𝜆𝑛 𝑥𝑛 𝑡 . Let R :  n e n .
Then 𝑅 = 𝑅(𝜆0 , 𝜆1 , … , 𝜆𝑛 ) is a solution of
𝑅/𝜆0
0=1−
𝑅/𝜆1
1−
𝑅/𝜆2
1−
⋱
1 − 𝑅/𝜆𝑛
Continued fractions are closely related
to tridiagonal matrices. This yields a
spectral representation of the mapping
(  0 ,  1 , ...,  n )  R .
54
Spectral Analysis*
Theorem: Consider the (n+2)x(n+2) symmetric,
non-negative and irreducible tridiagonal matrix:
Denote its eigenvalues by
Then
1 / 2
 n 2  R
.
.
A spectral representation of (  0 ,  1 , ...,  n )  R .
55
Application 1: Concavity
Let R :  e denote the steady-state
translation rate.
n
n
Theorem: 𝑅 = 𝑅(𝜆0, 𝜆1, … , 𝜆𝑛 ) is a strictly
concave function.
56
Maximizing Translation Rate
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.
57
Maximizing Translation Rate*
M ax R (  0 , 1 ,...,  n )
S ub :  i  0
w 0  0  w1 1  ...  w n  n  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.
58
Maximizing Translation Rate
w 0  0  w1  1  b .
*
*
59
Application 2: Sensitivity
Sensitivity of R to small changes in
the rates -> an eigenvalue sensitivity
problem.
60
Application 2: Sensitivity*
Theorem: Suppose that
log(𝜕𝑅/𝜕𝜆𝑖 )
𝜆0 =𝜆1 = ⋯ = 𝜆𝑛 .
Then
R
i
sin (

i 1
n3
 ) sin (
2 ( n  3 ) cos (
3
i2
n3

n3
)
.
)
Rates at the center of the chain are more
important.
*Poker, M. and Tuller, “Sensitivity of mRNA translation,
submitted, 2014.
61
Further Research
1. Analysis: controllability and
observability, stochastic rates, networks
of RFMs,…
2. Modifying the RFM (extended objects,
ribosome drop-off).
3. TASEP has been used to model:
biological motors, surface growth, traffic
flow, ants moving along a trail, Wi-Fi
networks,….
62
Conclusions
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!
63