Mathematical Identification of Critical Reactions in the Interlocked Feedback Model

Dynamic simulations are necessary for understanding the mechanism of how biochemical networks generate robust properties to environmental stresses or genetic changes. Sensitivity analysis allows the linking of robustness to network structure. However, it yields only local properties regarding a particular choice of plausible parameter values, because it is hard to know the exact parameter values in vivo. Global and firm results are needed that do not depend on particular parameter values. We propose mathematical analysis for robustness (MAR) that consists of the novel evolutionary search that explores all possible solution vectors of kinetic parameters satisfying the target dynamics and robustness analysis. New criteria, parameter spectrum width and the variability of solution vectors for parameters, are introduced to determine whether the search is exhaustive. In robustness analysis, in addition to single parameter sensitivity analysis, robustness to multiple parameter perturbation is defined. Combining the sensitivity analysis and the robustness analysis to multiple parameter perturbation enables identifying critical reactions. Use of MAR clearly identified the critical reactions responsible for determining the circadian cycle in the Drosophila interlocked circadian clock model. In highly robust models, while the parameter vectors are greatly varied, the critical reactions with a high sensitivity are uniquely determined. Interestingly, not only the per-tim loop but also the dclk-cyc loop strongly affect the period of PER, although the dclk-cyc loop hardly changes its amplitude and it is not potentially influential. In conclusion, MAR is a powerful method to explore wide parameter space without human-biases and to link a robust property to network architectures without knowing the exact parameter values. MAR identifies the reactions critically responsible for determining the period and amplitude in the interlocked feedback model and suggests that the circadian clock intensively evolves or designs the kinetic parameters so that it creates a highly robust cycle.

为阐明生化网络如何针对环境胁迫或遗传改变产生稳健特性的机制，动态模拟是必不可少的研究手段。敏感性分析可将稳健性与网络结构建立关联，但由于难以获取活体内的确切参数取值，该方法仅能针对特定的合理参数组合得到局部特性。因此，亟需不依赖特定参数取值的全局性、确定性分析结果。为此，我们提出了稳健性数学分析方法（Mathematical Analysis for Robustness，简称MAR），其包含两大核心模块：一是探索所有满足目标动力学特性的动力学参数解向量的新型进化搜索算法，二是稳健性分析环节。我们引入参数谱宽度与参数解向量变异性两项新准则，用于判断搜索过程是否穷尽了全部可行解空间。在稳健性分析中，除单参数敏感性分析外，我们还定义了多参数扰动下的系统稳健性。将敏感性分析与多参数扰动稳健性分析相结合，即可精准识别出网络中的关键反应节点。将MAR应用于果蝇连锁昼夜节律时钟模型后，清晰识别出了调控昼夜节律周期的关键反应节点。在高度稳健的模型中，尽管参数向量存在大幅波动，但具有高敏感性的关键反应节点仍可被唯一确定。值得注意的是，不仅per-tim环路，dclk-cyc环路也会对PER的周期产生显著影响——尽管dclk-cyc环路几乎不会改变其振幅，且此前被认为不具备潜在调控影响力。综上，MAR是一种无需人类主观偏见即可探索广阔参数空间的有效方法，可在未知确切参数值的前提下，将稳健特性与网络架构建立关联。MAR可在连锁反馈模型中识别出调控周期与振幅的关键反应节点，并表明昼夜节律时钟通过精细化进化或设计动力学参数，从而构建出高度稳健的节律周期。