TY - GEN

T1 - Mediating Ribosomal Competition by Splitting Pools

AU - Miller, Jared

AU - Al-Radhawi, M. Ali

AU - Sontag, Eduardo D.

N1 - Funding Information:
To prove non-degeneracy, it has been shown in [14], [13] that the Jacobian J∑of (13) at any point in the space can be written as J = ℓ=1LρℓQℓ for some ρℓ ≥ 0, rank-one matrices Qℓ and some L. Here, L = M (2 + s(2n + 1)). The coefficients {ρℓ} correspond to the partial derivatives of the rates {Rjpi}, {Rp±}, while {Qℓ} correspond to the network structure and are independent of the rates. Furthermore, it has been shown also [13] that the existence of an RLF implies that it sufficient to find one positive point ρ∗ = (ρ1∗, .., ρL∗ ) for which the (reduced) Jacobian is non-degenerate to show that is non-degenerate for all ρℓ > 0. We will find that point next by studying the structure of the Jacobian. Similar to a single pool [11, SI], it can be easily seen that J has non-negative off-diagonals and strictly negative diagonals (i.e, J is Metzler). In addition, conservation of the number of ribosomes implies that 1TJ = 0. By the definition of the Jacobian, all the entries in each column contain only the partial derivatives with respect to the state variable associated to the column. Hence, we can choose the corresponding ρℓ’s such the diagonal entry in each column is scaled to −1. Therefore, we consider the Jacobian evaluated at the chosen point ρ∗ such that J∗ = P −I, where I is the identity matrix and P a nonnegative irreducible column-stochastic matrix. By Perron-Frobenius Theorem, P has a maximal eigenvalue 1 with algebraic multiplicity 1. Therefore, J∗ has a single eigenvalue at 0 and the remaining eigenvalues have strictly negative real-parts. Hence, the reduced Jacobian at ρ∗ is nondegenerate. Robust non-degeneracy and GAS follows. The existence of a steady-state follows from Brouwer’s fixed point theorem since (13) evolves in a compact space (for a fixed Nr > 0), and uniqueness follows from non-degeneracy and GAS. The positivity of the steady-state follows from persistence of the ORFM which can be shown graphically by the absence of critical siphons [19]. ■ Acknowledgements: This research was partially funded by NSF grants 1849588 and 1716623. Jared Miller thanks Prof. Mario Sznaier for his support and discussions.
Publisher Copyright:
© 2021 American Automatic Control Council.

PY - 2021/5/25

Y1 - 2021/5/25

N2 - Synthetic biology constructs often rely upon the introduction of 'circuit' genes into host cells, in order to express novel proteins and thus endow the host with a desired behavior. The expression of these new genes 'consumes' existing resources in the cell, such as ATP, RNA polymerase, amino acids, and ribosomes. Ribosomal competition among strands of mRNA may be described by a system of nonlinear ODEs called the Ribosomal Flow Model (RFM). The competition for resources between host and circuit genes can be ameliorated by splitting the ribosome pool by use of orthogonal ribosomes, where the circuit genes are exclusively translated by mutated ribosomes. In this work, the RFM system is extended to include orthogonal ribosome competition. This Orthogonal Ribosomal Flow Model (ORFM) is proven to be stable through the use of Robust Lyapunov Functions. The optimization problem of maximizing the weighted protein translation rate by adjusting allocation of ribosomal species is formulated.

AB - Synthetic biology constructs often rely upon the introduction of 'circuit' genes into host cells, in order to express novel proteins and thus endow the host with a desired behavior. The expression of these new genes 'consumes' existing resources in the cell, such as ATP, RNA polymerase, amino acids, and ribosomes. Ribosomal competition among strands of mRNA may be described by a system of nonlinear ODEs called the Ribosomal Flow Model (RFM). The competition for resources between host and circuit genes can be ameliorated by splitting the ribosome pool by use of orthogonal ribosomes, where the circuit genes are exclusively translated by mutated ribosomes. In this work, the RFM system is extended to include orthogonal ribosome competition. This Orthogonal Ribosomal Flow Model (ORFM) is proven to be stable through the use of Robust Lyapunov Functions. The optimization problem of maximizing the weighted protein translation rate by adjusting allocation of ribosomal species is formulated.

UR - http://www.scopus.com/inward/record.url?scp=85111931130&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85111931130&partnerID=8YFLogxK

U2 - 10.23919/ACC50511.2021.9483415

DO - 10.23919/ACC50511.2021.9483415

M3 - Conference contribution

AN - SCOPUS:85111931130

T3 - Proceedings of the American Control Conference

SP - 1897

EP - 1902

BT - 2021 American Control Conference, ACC 2021

PB - Institute of Electrical and Electronics Engineers Inc.

T2 - 2021 American Control Conference, ACC 2021

Y2 - 25 May 2021 through 28 May 2021

ER -