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Transcript of 1 The Ribosome Flow Model Michael Margaliot School of Elec. Eng. Tel Aviv University, Israel Tamir...
1
The Ribosome Flow Model
Michael Margaliot
School of Elec. Eng.
Tel Aviv University, Israel
Tamir Tuller (Tel Aviv University)
Eduardo D. Sontag (Rutgers University)
Joint work with:
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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 FlowDuring 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)
Particles can only hop to empty sites (SE)
Movement is unidirectional (TA)
A stochastic model: particles hop along a lattice of consecutive sites
Simulating TASEP
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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*
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*Schadschneider, Chowdhury & Nishinari, Stochastic Transport in Complex Systems: From Molecules to Vehicles, 2010.
1. Mean field approximations
2. Bethe ansatz
Ribosome Flow Model*
*Reuveni, Meilijson, Kupiec, Ruppin & Tuller, “Genome-scale analysis of translation elongation with a ribosome flow model”, PLoS Comput. Biol., 2011 9
A 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.
Ribosome Flow Model
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(normalized) number of ribosomes at site i
State-variables:
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.
Simulation Results
( ) | ( ; ) | .fJ u x t u
0(0) .x x
All trajectories emanating from
remain in , and converge to a unique
equilibrium point e. 13
0.ft e
Analysis of the RFM
Uses tools from:
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Contraction theory
Monotone systems theory
Analytic theory of continued fractions
Contraction Theory*
The system:
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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.
Contraction Theory
Trajectories contract to each other at
an exponential rate.16
a
b
x(t,0,a)
x(t,0,b)
Implications of Contraction
1. Trajectories converge to a unique
equilibrium point;
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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
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
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This suggests that in order to prove
contraction we need to (uniformly)
bound J(x).
How to Prove Contraction?
Let be a vector norm.
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The induced matrix norm
is:
The induced matrix measure
is:
How to Prove Contraction?
Intuition on the matrix measure:
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Consider Then to 1st order in
so
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.
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.”
RFM is not Contracting on C
For n=3:
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so for is singular
and thus not Hurwitz.
Contraction After a Short Transient (CAST)*
Definition is a CAST if
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*M., Sontag & Tuller, “Entrainment to Periodic Initiation and Transition Rates in the Ribosome Flow Model”, submitted, 2013 .
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.& 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.
Entrainment in the RFM
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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 & Tuller, “Entrainment to Periodic Initiation and Transition Rates in the Ribosome Flow Model”, submitted, 2013 .
Entrainment in the RFM
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Here n=3,
Analysis of the RFM
Uses tools from:
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Contraction theory
Monotone systems theory
Analytic theory of continued fractions
Monotone Dynamical Systems*Define a (partial) ordering between vectors
in Rn by:
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*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.
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.
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*Sontag, “Monotone and Near-Monotone Biochemical Networks”, Systems & Synthetic Biology, 2007
When is a System Monotone?
Theorem (Kamke Condition.) Suppose
that f satisfies:
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then is monotone.
Intuition: assume monotonicity is lost,
then
Verifying the Kamke Condition
Theorem cooperativity Kamke
condition ( system is monotone)
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This means that increasing increases
Definition is called cooperative if
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:
RFM is Cooperative
increase. A “traffic jam” in a site induces
“traffic jams” in the neighboring sites.
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Intuition if x2 increases then
and
RFM is Monotone
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Biological implication: a larger initial
distribution of ribosomes induces a
larger distribution of ribosomes for all
time.
Analysis of the RFM
Uses tools from:
40
Contraction theory
Monotone systems theory
Analytic theory of continued fractions
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Continued FractionsSuppose (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 th–order polynomial equation in en.
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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 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,
*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.
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.*
RFM with Feedback*
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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.
RFM with Feedback*
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Theorem In the homogeneous case,
where
Biological implication: may be useful,
perhaps, for re-engineering gene translation.
Further Research
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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,….
Summary
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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!