Self-assembly-based posttranslational protein oscillators

Abstract

Recent advances in synthetic posttranslational protein circuits are substantially impacting the landscape of cellular engineering and offer several advantages compared to traditional gene circuits. However, engineering dynamic phenomena such as oscillations in protein-level circuits remains an outstanding challenge. Few examples of biological posttranslational oscillators are known, necessitating theoretical progress to determine realizable oscillators. We construct mathematical models for two posttranslational oscillators, using few components that interact only through reversible binding and phosphorylation/dephosphorylation reactions. Our designed oscillators rely on the self-assembly of two protein species into multimeric functional enzymes that respectively inhibit and enhance this self-assembly. We limit our analysis to within experimental constraints, finding (i) significant portions of the restricted parameter space yielding oscillations and (ii) that oscillation periods can be tuned by several orders of magnitude using recent advances in computational protein design. Our work paves the way for the rational design and realization of protein-based dynamic systems.

Fig. 1. Bounded self-assembly oscillator.

(A) Oscillator topology. By phophorylating monomers, kinase multimers (red; top) inhibit their own and phosphatase multimer (blue; bottom) self-assembly. Similarly, phophatase multimers counteract this inhibition, as do constitutive phosphatases (center). (B) Bounded self-assembly reactions. Monomers contain two halves of a split enzyme: either kinase (red; top) or phosphatase (blue; bottom). Monomers can self-assemble into multimers of specified size (here, tetramers are pictured, corresponding to n = m = 4). Kinase (phosphatase) multimers can (de)phosphorylate the monomers. A constitutive phosphatase is also able to dephosphorylate the monomers (not pictured). Phosphorylated monomers cannot participate in the self-assembly. Reactions are shown in pictorial form above each corresponding chemical equation. The full set of differential equations corresponding to these reactions is given in eq. S1.

Fig. 2. Bounded self-assembly can yield oscillations using experimentally realizable parameters.

Numerical integration of eq. S1 displays parameter regimes leading to oscillations within experimental constraints. Each subplot shows the location of oscillating parameter sets as a function of k_dκ and k_dρ for given k_bκ and k_bρ; the latter two are varied for each subplot. Aside from experimental constraints (see main text for discussion), we set n = m = 2, κ_tot = ρ_tot = 10 μM, and ${\tilde{P}}_{\text{tot}}={10}^{-4}$ μM. Blue points denote parameter sets leading to sustained oscillations; yellow points denote parameter sets leading to steady state. To the right of the plot, we show a few example trajectories in K-P phase space. The closed trajectories correspond to sustained oscillations; the final trajectory, the spiral, corresponds to a decaying oscillation and therefore to a yellow point in the figure.

Fig. 3. Analytical results for bounded self-assembly oscillator.

In this figure, we show results from the analytically simplified bounded self-assembly oscillator (Eq. 1), using n = m = 2. (A) Oscillation schematic. We visualize a sample oscillation using randomly and arbitrarily chosen parameters satisfying experimental constraints. Oscillations require phosphatase multimers (blue) to dissociate faster than kinase multimers (red). The system starts with self-assembled kinases and phosphatases (top right). After rapid phosphatase disassembly and phosphorylation by the kinase multimers (bottom right), the kinases slowly disassemble, which enables the gradual dephosphorylation and self-assembly of the phosphatase monomers (bottom left). The assembled phosphatases are then able to rapidly promote their own and kinase self-assembly through dephosphorylation, returning the system to its initial state (top right). (B) Onset of oscillations. Numerical integration demonstrates consistency with Eq. 5 for the appearance of oscillations in the appropriate limits. Each point represents a random set of parameters, sampled within the experimentally realizable limits as described in the main text. Oscillating (blue) and non-oscillating (yellow) parameter sets can be well-separated by dimensionless combinations of parameters γ and ν. Dashed lines show where the dimensionless parameters on the axes equal unity. (C) Oscillation frequency. Intuition from linear stability analysis of the fixed point suggests that for the n = m = 2 system considered numerically, oscillation frequency may be determined by the dissociation rate of kinase multimers, $k_{\mathrm{u}\mathrm{\kappa }}$. Numerical integration demonstrates that $k_{\mathrm{u}\mathrm{\kappa }}$ is indeed highly predictive of oscillation frequency (R² ≈ 0.66; R² ≈ 0.93 in log space) and underestimates the true frequency by a typical factor of ~4. Black dashed line shows ω = k_uκ.

Fig. 4. Unbounded self-assembly oscillator.

(A) Unbounded self-assembly reactions. We consider a related system to that shown in Fig. 1 but relying on kinase and phosphatase monomers that self-assemble into unbounded fibers of arbitrary length. In addition, we assume that the final monomer of each fiber can get phosphorylated by a kinase multimer, at which point it can no longer rejoin the fiber until it is dephosphorylated. (B) Unbounded self-assembly oscillations using experimentally realizable parameters. Numerical integration of Eq. 9 displays parameter regimes leading to oscillations within experimental constraints. Eq. 9 was used in place of the full system of equations (eq. S8) because of the infinite dimensionality of the latter. ρ_tot sets the concentration scale.