Lohmann Johannes, D'Huys Otti, Haynes Nicholas D, Schöll Eckehard, Gauthier Daniel J
Department of Physics, Duke University, Durham, North Carolina 27708, USA.
Institut für Theoretische Physik, Technische Universität Berlin, 10623 Berlin, Germany.
Phys Rev E. 2017 Feb;95(2-1):022211. doi: 10.1103/PhysRevE.95.022211. Epub 2017 Feb 17.
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.
具有开关样相互作用的生化系统，如基因调控网络，可以通过自主布尔网络得到很好的建模。具体而言，基因相互作用的拓扑结构和逻辑可以用连续分段线性微分方程系统来描述，从而能够对特定网络的动态进行分析预测。然而，大多数模型没有考虑与空间运输、mRNA转录和翻译相关的链接上的时间延迟。为了解决这个问题，我们开发了一个实验平台，以实现一个具有三个抑制节点的时间延迟自主布尔网络，即阻遏振荡器，并利用它来研究随着链接上的时间延迟变化而产生的动态。我们观察到各种具有极长寿命的近周期振荡瞬态模式，这些模式由于延迟而在小网络基序中出现，并且与最终渐近稳定的周期吸引子不同。对于给定网络的重复实验，我们发现随机过程会导致瞬态时间的广泛分布，其尾部呈指数分布。在某些情况下，瞬态时间非常长，以至于在具有有限寿命的生物系统中，是否能接近吸引子都值得怀疑。为了抵消长瞬态，我们通过实验表明，偶尔对时间延迟施加小的扰动可以迫使轨迹迅速接近吸引子。
