Cost-effective circadian mechanism: rhythmic degradation of circadian proteins spontaneously emerges without rhythmic post-translational regulation

Circadian protein oscillations are maintained by the lifelong repetition of protein production and degradation in daily balance. It comes at the cost of ever-replayed, futile protein synthesis each day. This biosynthetic cost with a given oscillatory protein profile is relievable by a rhythmic, not constant, degradation rate that selectively peaks at the right time of day but remains low elsewhere, saving much of the gross protein loss and of the replenishing protein synthesis. Here, our mathematical modeling reveals that the rhythmic degradation rate of proteins with circadian production spontaneously emerges under steady and limited activity of proteolytic mediators and does not necessarily require rhythmic post-translational regulation of previous focus. Additional (yet steady) post-translational modifications in a proteolytic pathway can further facilitate the degradation's rhythmicity in favor of the biosynthetic cost saving. Our work is supported by animal and plant circadian data, offering a generic mechanism for potentially widespread, time-dependent protein turnover

Rhythmic Protein Degradation for Cost-Effective Circadian Oscillation

Circadian protein oscillations are maintained by the lifelong repetition of protein production and degradation in daily balance. It comes at the cost of ever-replayed, futile protein synthesis each day. This biosynthetic cost with a given oscillatory protein profile is relievable by a rhythmic, not constant, degradation rate that selectively peaks at the right time of day but remains low elsewhere, saving much of the gross protein loss and of the replenishing protein synthesis. Here, our mathematical modeling reveals that the rhythmic degradation rate of proteins with circadian production spontaneously emerges under steady and limited activity of proteolytic mediators and does not necessarily require rhythmic post-translational regulation of previous focus. Additional (yet steady) post-translational modifications in a proteolytic pathway can further facilitate the degradation???s rhythmicity in favor of the biosynthetic cost saving. Our work is supported by animal and plant circadian data, offering a generic mechanism for potentially wide-spread, time-dependent protein turnover

Backward simulation for inferring hidden biomolecular kinetic profiles

Our backward simulation (BS) is an approach to infer the dynamics of individual components in ordinary differential equation (ODE) models, given the information on relatively downstream components or their sums. Here, we demonstrate the use of BS to infer protein synthesis rates with a given profile of protein concentrations over time in a circadian system. This protocol can also be applied to a wide range of problems with undetermined dynamics at the upstream levels. For complete details on the use and execution of this protocol, please refer to Lim et al. (2021)

Positive autoregulation and induction kinetics.

(a) Protein production mechanism with positive autoregulation in the presence of inducers. (b) Bifurcation diagram of the simulated protein level as a function of η (proxy for an inducer level). The steady state is plotted as η increases (solid line) or decreases (dashed line). Acute induction can be simulated by a sudden change of η = 0 to η>ηc in the shaded area. (c) Time-series of protein levels from the full, ETS, and QSSA models upon acute induction at time 0 h with η = 2.42 (left) or η = 200 (right). (d) The full model-to-QSSA difference in response time as a function of . Both the simulated and analytically-estimated differences are presented. The analytical estimation is based on Eq (9). For more details of (b)–(d), refer to Text I and Tables H and I in S1 Appendix.</p

S1 Appendix -

Texts A–L, Fig A–H, and Tables A–R. Text A. Rate law overview and derivation. Text B. Amplitude overestimation with Eq (S14). Text C. Amplitude overestimation with simpler new rate laws. Text D. Rate law derivation for TF–DNA interactions. Text E. Preconditions of rate laws. Text F. Metabolic reaction and transport kinetics. Text G. Protein–protein interaction. Text H. TF–DNA interaction. Text I. Positive autogenous control. Text J. Negative autogenous control. Text K. Rhythmic protein degradation. Text L. Simulation and analysis methods. Fig A. Preconditions of rate laws. Fig B. Oxaloacetate (substrate) conversion by malate dehydrogenase (enzyme). Fig C. Protein–protein interaction modeling. Fig D. Protein ZTL–GI interaction in Arabidopsis. Fig E. TF–DNA interaction modeling. Fig F. Phase portrait of induction kinetics with η>ηc in the case of positive autoregulation. Fig G. Negative autoregulation and induction kinetics. Fig H. The sQSSA- and tQSSA-based parameter estimation. Table A. Enzyme–substrate pairs of metabolic reactions in E. coli (refer to Text F). Table B. The PTS system of E. coli (refer to Text F). Table C. Parameter ranges of protein–protein and TF–DNA interaction models (refer to Texts G and H). Table D. Parameter ranges for ZTL profile simulation (refer to Text G). Table E. Parameter ranges for induction kinetics simulation [refer to Texts I and J (associated with Section Autogenous control in the main text)]. Table F. Parameter ranges for protein degradation simulation [refer to Text K (associated with Section Rhythmic degradation of circadian proteins in the main text)]. Table G. Parameters used in Fig 1(A) and 1(B). Table H. Parameters used in Fig 2(B)–2(D) and F. Table I. Simulated values of the full model-to-QSSA difference in Fig 2(D). Table J. Parameters used in Fig 3(C). Table K. Parameters used in Fig A(a) and A(b). Table L. Parameters used in Fig C(a). Table M. Parameters used in Fig C(f). Table N. Parameters used in Fig D(a) and D(b). Table O. Parameters used in Fig E(a). Table P. Parameters used in Fig E(d). Table Q. Parameters used in Fig G(b)–G(e). Table R. Simulated values of the QSSA-to-full model difference in Fig G(e). (PDF)</p

Generalization of the MM rate law for time-varying molecular concentrations, referred to as the ETS.

(a) Two different molecules A and B bind to each other and form their complex. (b) A TF binds to a DNA molecule to regulate mRNA expression (RNA polymerase and other molecules are omitted here). In (a) and (b), the graphs show the comparison among the exact time-course profile of the complex concentration, the tQSSA-based (a) or QSSA-based (b) profile, and the ETS-based profile. The relationship between the tQSSA (or QSSA) and the ETS is illustrated through the effective time delay in the ETS. Notations ka, kd, kδ, t, ΔtQ(t), K, and ATF(t) are defined in the description of Eqs (1)–(3) and (6)–(8). Simulations in (a) and (b) are based on periodic oscillation models in Texts G and H in S1 Appendix, respectively, with their parameters in Table G in S1 Appendix.</p

Parameter estimation for protein–protein and TF–DNA interaction models.

(a) The scatter plot of the relative errors of the tQSSA- and ETS-estimated K values for a protein–protein interaction model. (b) The scatter plot of the relative errors of the QSSA- and ETS-estimated K values for a TF–DNA interaction model. In (a) and (b), a diagonal line corresponds to the cases where the two estimates have the same relative errors. (c) The probability distribution of the relative error of the ETS-estimated kδ for the protein–protein interaction model in (a). (d) The probability distribution of the relative error of the ETS-estimated kδ for the TF–DNA interaction model in (b). Regarding (a)–(d), a subset of simulated conditions gave relative errors outside the presented ranges here, but they did not alter the observed patterns. For more details, refer to Text L in S1 Appendix.</p

Rhythmic degradation of circadian proteins.

(a) The experimental abundance levels (solid line) and degradation rates (open circles) of the mouse PERIOD2 (PER2) protein [43]. (b) The experimental abundance levels (dots, interpolated by a solid line) and degradation rates (open circles) of PSEUDO RESPONSE REGULATOR 7 (PRR7) protein in Arabidopsis thaliana [44,45,48]. Horizontal white and black segments correspond to light and dark intervals, respectively. (c) A simulated protein degradation rate from the full kinetic model and its ETS- and QSSA-based estimates, when the degradation depends on a single PTM. In addition, the protein abundance profile is presented here (gray solid line). A vertical dashed line corresponds to the peak time of −A′(t)/A(t) where A(t) is a protein abundance. The parameters are provided in Table J in S1 Appendix. (d) The probability distribution of the peak-time difference between a degradation rate and −A′(t)/A(t) for each number of PTMs (n) required for the degradation. The probability distribution was obtained with randomly-sampled parameter sets in Table F in S1 Appendix. (e) The probability distribution of the relative amplitude of a simulated degradation rate (top) or its estimate in Eq (13) (bottom) for each n, when the relative amplitude of a protein abundance is 1. (f) The probability distribution of the ratio of the simulated to estimated relative amplitude of a degradation rate for each n. For more details of (a)–(f), refer to Text K in S1 Appendix.</p