Temporal precision of molecular events with regulation and feedback

Cellular behaviors such as migration, division, and differentiation rely on precise timing, and yet the molecular events that govern these behaviors are highly stochastic. We investigate regulatory strategies that decrease the timing noise of molecular events. Autoregulatory feedback increases noise. Yet we find that in the presence of regulation by a second species, autoregulatory feedback decreases noise. To explain this finding, we develop a method to calculate the optimal regulation function that minimizes the timing noise. The method reveals that the combination of feedback and regulation minimizes noise by maximizing the number of molecular events that must happen in sequence before a threshold is crossed. We compute the optimal timing precision for all two-node networks with regulation and feedback, derive a generic lower bound on timing noise, and discuss our results in the context of neuroblast migration during Caenorhabditis elegans development.

Figure 1

Feedback increases timing precision in the presence but not absence of regulation. (a) A species $Y$ crosses a molecule-number threshold $y_{*}$ at mean time $t_{*}$ with timing variance ${\sigma }_t^2$. (b) Feedback increases the variance. However, in the presence of regulation by a second species $X$, feedback on either (c) $Y$ or (d) $X$ can decrease the variance. Parameters are $K_y=2.5$ in (b); ${\alpha }_0{t}_{*}=10,H_x=-0.5,H_{xy}=-H_y,K_x=15,K_y=5$, and $K_{xy}=6$ in (c); ${\alpha }_0{t}_{*}=10,H_y=4,K_x=10$, and $K_y=7.5$ in (d); and $y_{*}=10$ throughout. (a) is generated by Gillespie's simulation [14]; (b)–(d) are generated by solving the master equation numerically [9].

Figure 2

Optimal regulation functions that minimize timing variance. (a) Without feedback, $X$ activates $Y$, (b) allowing $\overline{y}$ to accelerate before crossing $y_{*}$. (c) With feedback on $Y,X$ acts as a “timer” for $Y$, allowing $Y$ to self-repress at early times and self-activate at late times and (d) providing further, late-phase acceleration of $\overline{y}$. (e) With feedback on $X$, it represses itself and activates $Y$ sharply, (f) resulting in kinked dynamics where $\overline{x}$ and $\overline{y}$ growth are separated in time. Parameters are ${\alpha }_0{t}_{*}=7$ in (a)–(d), $x_{*}=7$ in (e) and (f), and $y_{*}=10$ throughout.

Figure 3

Properties that minimize timing variance: (a) large path length $S$, (b) constant velocity $v\left(t\right)$ along the path, and (c) small path length variance ${\sigma }_S^2$. Parameters are as in Fig. 2.

Figure 4

(a) Ranking of timing variance for all one- and two-node networks. The global minimum is ${\sigma }_t^2{y}_{*}/t_{*}^2=y_{*}/(x_{*}+y_{*})$. In the red network, link 1, link 2, or both are required. Parameters are as in Fig. 2. (b) Mean dynamics and regulation function (inset) for the case when $X$ is degraded. Here ${\alpha }_0{t}_{*}=3.5$.