Transient dynamics and their control in time-delay autonomous Boolean ring networks

Biochemical systems with switch-like interactions, such as gene regulatory networks, are well modeled by autonomous Boolean networks. Specifically, the topology and logic of gene interactions can be described by systems of continuous piecewise-linear differential equations, enabling analytical predictions of the dynamics of specific networks. However, most models do not account for time delays along links associated with spatial transport, mRNA transcription, and translation. To address this issue, we have developed an experimental test bed to realize a time-delay autonomous Boolean network with three inhibitory nodes, known as a repressilator, and use it to study the dynamics that arise as time delays along the links vary. We observe various nearly periodic oscillatory transient patterns with extremely long lifetime, which emerge in small network motifs due to the delay, and which are distinct from the eventual asymptotically stable periodic attractors. For repeated experiments with a given network, we find that stochastic processes give rise to a broad distribution of transient times with an exponential tail. In some cases, the transients are so long that it is doubtful the attractors will ever be approached in a biological system that has a finite lifetime. To counteract the long transients, we show experimentally that small, occasional perturbations applied to the time delays can force the trajectories to rapidly approach the attractors.

Figure 1

(a) Ring network of $N=3$ nodes with inhibitory interactions. (b) Description of panel (a) as time-delay autonomous Boolean network. (c) State transition diagram of the autonomous Boolean network (b) without link delays.

Figure 2

Temporal evolution of the transient dynamics. (a) Waveform of one node in a network with $\mathbf{n}=(2,2,2)$ delay elements initialized in the (1,1,1) state. The initial nearly periodic in-phase behavior eventually gives way to out-of-phase oscillations after $T\sim 4.05\mu \mathrm{s}$. (b) Corresponding three-dimensional phase portrait of the trajectory, which evolves from in-phase nearly periodic oscillations (left) to the asymptotic out-of-phase periodic attractor (right).

Figure 3

Mean transient durations for networks with a homogeneous number of $n$ delay elements in each network link, repeatedly initialized in the (0,0,0) state. For $n=5$, the transients for each experimental run last longer than 40 min at which point the experiment terminated. Here, the data point only signifies the termination point and the arrow indicates that the mean is longer than this value. For each data point in the white region, we collect 100 000 samples. In the gray region, we conduct 30 experimental runs for $n=5$, 32 000 for $n=6$, and 94 669 for $n=7$. The error bars indicate the standard deviation.

Figure 4

Probability density function for the transient durations of networks with (a) $\mathbf{n}=(2,2,2)$ and (b) $\mathbf{n}=(7,7,7)$ delay elements. Here, a histogram is generated indicating the number of instances that we observe a transient time within $T$ and $T+\mathrm{\Delta }T$, normalized by $\mathrm{\Delta }T$ and the total number of observed transient times. It approximates the transient time probability density $P\left(T\right)$, where $P\left(T\right)\mathrm{\Delta }T$ is the probability of observing a time interval $T$ when $\mathrm{\Delta }T$ is small. (b) The mean transient duration $\langle T\rangle$ is 0.41 s and the red line shows an exponential fit $[p\left(T\right)\propto e^{-\lambda T}]$ to the data with exponent $\lambda =2.44{\mathrm{s}}^{-1}$.

Figure 5

Phase portrait of two different transient trajectories of a network with $\mathbf{n}=(4,4,4)$ delay elements, both initialized in the (1,1,1) state.

Figure 6

Transient duration distributions in the presence of heterogeneities and active feedback control. (a) Transient duration distributions in a network with $\mathbf{n}=(7,7,7)$ delay elements when individually altering the timing of the initial conditions of each node by roughly 100 ps. The mean transient durations when delaying nodes 1, 2, and 3 are $\langle T\rangle =5.93$ ms, $\langle T\rangle =8.05$ ms, and $\langle T\rangle =151.46$ ms, respectively. We collected 30 000 samples for each distribution. (b) Transient duration distributions in a network with $\mathbf{n}=(6,6,6)$ delay elements, when perturbing the timing of a specific traveling edge. Note that the time scale for the distribution with no perturbation (red) is seconds.

Figure 7

Transient durations for the map Eqs. (5) and (6) for $N=3$ with no stochastic fluctuations in the time delay ($\xi =0$). (a) Transient durations as a function of the mean delay $\tau$ for a fixed heterogeneity $\mathrm{\Delta }=0.1$, defined by ${\tau }_1=\tau +\mathrm{\Delta }/2$, ${\tau }_2=\tau$, and ${\tau }_3=\tau -\mathrm{\Delta }/2$. Differences in initialization timing are not explicitly taken into account, i.e., ${\delta }_i=0$. An exponential fit ($T\propto e^{\lambda \tau }$) to the data with exponent $\lambda =1.17$ is shown in red. (b) Transient durations as a function of delay heterogeneity $\mathrm{\Delta }$ for a fixed mean delay $\tau =8$.

Figure 8

Transient durations for the map Eq. (6) for $N=3$. (a) Mean transient duration as a function of the mean delay $\tau$ for ${\sigma }^2={10}^{-4}$. (b) Probability density function of the transient duration for $\tau =10$ and ${\sigma }^2={10}^{-4}$.

Figure 9

Verilog code defining a ring network with $N=3$ inhibitory nodes and heterogeneous link time delays.

Figure 10

Scheme for perturbing a single Boolean transition every other round trip using a mod-3 asynchronous counter. The blue rectangles represent a NOT gate an the yellow rectangles represent the d-type flip-flops.