Kim Jae Kyoung, Sontag Eduardo D
Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon, Korea.
Department of Mathematics and Center for Quantitative Biology, Rutgers University, New Brunswick, New Jersey, United States of America.
PLoS Comput Biol. 2017 Jun 5;13(6):e1005571. doi: 10.1371/journal.pcbi.1005571. eCollection 2017 Jun.
Biochemical reaction networks (BRNs) in a cell frequently consist of reactions with disparate timescales. The stochastic simulations of such multiscale BRNs are prohibitively slow due to high computational cost for the simulations of fast reactions. One way to resolve this problem uses the fact that fast species regulated by fast reactions quickly equilibrate to their stationary distribution while slow species are unlikely to be changed. Thus, on a slow timescale, fast species can be replaced by their quasi-steady state (QSS): their stationary conditional expectation values for given slow species. As the QSS are determined solely by the state of slow species, such replacement leads to a reduced model, where fast species are eliminated. However, it is challenging to derive the QSS in the presence of nonlinear reactions. While various approximation schemes for the QSS have been developed, they often lead to considerable errors. Here, we propose two classes of multiscale BRNs which can be reduced by deriving an exact QSS rather than approximations. Specifically, if fast species constitute either a feedforward network or a complex balanced network, the reduced model based on the exact QSS can be derived. Such BRNs are frequently observed in a cell as the feedforward network is one of fundamental motifs of gene or protein regulatory networks. Furthermore, complex balanced networks also include various types of fast reversible bindings such as bindings between transcriptional factors and gene regulatory sites. The reduced models based on exact QSS, which can be calculated by the computational packages provided in this work, accurately approximate the slow scale dynamics of the original full model with much lower computational cost.
细胞中的生化反应网络（BRNs）通常由具有不同时间尺度的反应组成。由于快速反应模拟的计算成本很高，此类多尺度BRNs的随机模拟速度极其缓慢。解决这个问题的一种方法是利用这样一个事实：由快速反应调节的快速物种会迅速达到其稳态分布，而慢速物种不太可能发生变化。因此，在慢时间尺度上，快速物种可以用它们的准稳态（QSS）来代替：即给定慢速物种时它们的稳态条件期望值。由于QSS仅由慢速物种的状态决定，这种替换会导致一个简化模型，其中快速物种被消除。然而，在存在非线性反应的情况下推导QSS具有挑战性。虽然已经开发了各种QSS近似方案，但它们往往会导致相当大的误差。在这里，我们提出了两类多尺度BRNs，通过推导精确的QSS而不是近似值可以对其进行简化。具体来说，如果快速物种构成前馈网络或复杂平衡网络，就可以推导出基于精确QSS的简化模型。由于前馈网络是基因或蛋白质调控网络的基本基序之一，此类BRNs在细胞中经常出现。此外，复杂平衡网络还包括各种类型的快速可逆结合，例如转录因子与基因调控位点之间的结合。基于精确QSS的简化模型可以通过本工作提供的计算软件包来计算，能够以低得多的计算成本准确地近似原始完整模型的慢尺度动态。
