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)
1
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
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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
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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
only bind if the site is empty.
, but can
*Reuveni, Meilijson, Kupiec, Ruppin & Tuller,
“Genome-scale analysis of translation elongation
with a ribosome flow model”, PLoS Comput. Biol.,
2011
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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)  x0 .
t f  0.
J (u) e| x(t f ; u) |.
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.
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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 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.
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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 1st 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 L1 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
is singular
and thus not Hurwitz.
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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.
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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
.
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; not affected by perturbations.
*M.& Tuller, “Stability Analysis of the Ribosome
Flow Model”, IEEE TCBB, 2012.
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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.
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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 Rn 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
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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.
*Sontag, “Monotone and Near-Monotone Biochemical
Networks”, Systems & Synthetic Biology, 2007
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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 nonnegative on C. Thus, the RFM is a
cooperative system.
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RFM is Cooperative
Intuition if x2 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 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
polynomial equation in en.
th–order
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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”, IEEE TCBB,
2012
46
mRNA Circularization*
*Craig, Haghighat, Yu & Sonenberg, ”Interaction of
Polyadenylate-Binding Protein with the eIF4G homologue PAIP
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enhances translation”, Nature, 1998
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
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TAC, 2003
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
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RFM with Feedback*
Theorem In the homogeneous case,
where
Biological implication: may be useful,
perhaps, for re-engineering gene translation.50
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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