1 The Ribosome Flow Model Michael Margaliot School of Electrical Engineering Tel Aviv University, Israel Tamir Tuller (Tel Aviv University) Eduardo D. Sontag (Rutgers University) Joint work with: Gilad Poker Yoram Zarai

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 implications

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 Gene (DNA) Transcription mRNA Translation Protein 4

Gene Expression 5

Translation 6

Translation: the Genetic Code The mRNA is built of codons 7

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 8

Flow of Ribosomes 9 Source:

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, * Ingolia, Nature Reviews Genetics,2014.

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, Spitzer, Adv. Math., *Zia, Dong & Schmittmann, “Modeling Translation in Protein Synthesis with TASEP: A Tutorial and Recent Developments”, J Stat Phys, 2010

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

Ribosome Flow Model (RFM)* Transition rates:. = initiation rate State 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 codons

14 Ribosome Flow Model unidirectional movement & simple exclusion

15 Ribosome Flow Model is the translation rate at time

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

Contraction Theory* The system: 17 is contractive on a convex set K, with contraction rate c>0, if for all * Lohmiller & Slotine, “On Contraction Analysis for Nonlinear Systems”, Automatica, *Aminzare & Sontag, “Contraction methods for nonlinear systems: a brief introduction and some open problems”, IEEE CDC 2014.

Contraction Theory Trajectories contract to each other at an exponential rate. 18 a b x(t,0,a) x(t,0,b)

Implications of Contraction 1. Trajectories converge to a unique equilibrium point (if one exists); The system entrains to periodic excitations.

Contraction and Entrainment* Definition: is T-periodic if 20 *Russo, di Bernardo & Sontag, “Global Entrainment of Transcriptional Systems to Periodic Inputs”, PLoS Comput. Biol., Theorem : The contracting and T-periodic system admits a unique periodic solution of period T, and

Proving Contraction The Jacobian of is the nxn matrix 21

Proving Contraction The infinitesimal distance between trajectories evolves according to 22 This suggests that in order to prove contraction we need to (uniformly) bound J(x).

Proving Contraction Let be a vector norm. 23 The induced matrix norm is: The induced matrix measure is:

Proving Contraction Intuition on the matrix measure: 24 Consider Then to 1 st order in so

Proving Contraction Theorem: Consider the system 25 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, 26 and for the matrix measure induced by the L 1 vector norm: for all The RFM is on the “verge of contraction.”

RFM is not Contracting on C For n=3: 27 so for is singular and thus not Hurwitz.

Contraction After a Short Transient (CAST)* Definition: is CAST if 28 *Sontag, M., and Tuller, “On three generalizations of contraction”, IEEE CDC 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). 29

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, Biological interpretation: the parameters determine a unique steady-state of ribosome distributions and synthesis rate.

Simulation Results All trajectories emanating from C=[0,1] 3 remain in C, and converge to a unique equilibrium point e. 31 e

Entrainment in the RFM 32

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.

Entrainment in the RFM 34 Here n=3,

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,…

Monotone Dynamical Systems* Define a (partial) ordering between vectors in R n 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.

Monotone Systems in the Life Sciences* -behavior is ordered and robust with respect to parameter values -large 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.

When is a System Monotone? Theorem: cooperativity system is monotone 38 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. 39 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. 40 Intuition: if x 2 increases then and

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)=b 1 x 2 (0)=b 2, … a≤b x(t,a)≤x(t,b) x 1 (t,b) x 2 (t,b),… x 1 (t,a) x 2 (t,a),…

Analysis of the RFM 42 Contraction theory Monotone systems theory Analytic theory of continued fractions Spectral analysis Convex optimization theory Random matrix theory,…

43 Continued Fractions Suppose (for simplicity) that n =3. Then Let denote the unique equilibrium point in C. Then

44 Continued Fractions This yields: Every e i can be expressed as a continued fraction of e 3.

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.

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.

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.

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).

mRNA Circularization* 49 * Craig, Haghighat, Yu & Sonenberg, ”Interaction of Polyadenylate-Binding Protein with the eIF4G homologue PAIP enhances translation”, Nature, 1998

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.*

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.

HRFM with Feedback 52 Theorem: In the homogeneous case, where. Biological implication: may be useful, perhaps, for re-engineering gene translation.

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,…

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 mapping

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

Application 1: Concavity 56 Let denote the steady-state translation rate. Theorem: is a strictly concave function.

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.

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.

Maximizing Translation Rate 59

Application 2: Sensitivity 60 Sensitivity of R to small changes in the rates -> an eigenvalue sensitivity problem.

Application 2: Sensitivity* 61 Theorem: Suppose that *Poker, M. and Tuller, “Sensitivity of mRNA translation, submitted, Then Rates at the center of the chain are more important.

Further Research 62 1.Analysis: controllability and observability, 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).

Conclusions 63 The Ribosome Flow Model is: (1) useful; (2) amenable to analysis. Papers available on-line at: Recently developed techniques provide more and more data on the translation process. Computational models are thus becoming more and more important. THANK YOU!