Ghasemi Omid, Lindsey Merry L, Yang Tianyi, Nguyen Nguyen, Huang Yufei, Jin Yu-Fang
Department of Electrical and Computer Engineering, University of Texas at San Antonio, San Antonio, TX, USA.
BMC Syst Biol. 2011;5 Suppl 3(Suppl 3):S9. doi: 10.1186/1752-0509-5-S3-S9. Epub 2011 Dec 23.
The availability of temporal measurements on biological experiments has significantly promoted research areas in systems biology. To gain insight into the interaction and regulation of biological systems, mathematical frameworks such as ordinary differential equations have been widely applied to model biological pathways and interpret the temporal data. Hill equations are the preferred formats to represent the reaction rate in differential equation frameworks, due to their simple structures and their capabilities for easy fitting to saturated experimental measurements. However, Hill equations are highly nonlinearly parameterized functions, and parameters in these functions cannot be measured easily. Additionally, because of its high nonlinearity, adaptive parameter estimation algorithms developed for linear parameterized differential equations cannot be applied. Therefore, parameter estimation in nonlinearly parameterized differential equation models for biological pathways is both challenging and rewarding. In this study, we propose a Bayesian parameter estimation algorithm to estimate parameters in nonlinear mathematical models for biological pathways using time series data.
We used the Runge-Kutta method to transform differential equations to difference equations assuming a known structure of the differential equations. This transformation allowed us to generate predictions dependent on previous states and to apply a Bayesian approach, namely, the Markov chain Monte Carlo (MCMC) method. We applied this approach to the biological pathways involved in the left ventricle (LV) response to myocardial infarction (MI) and verified our algorithm by estimating two parameters in a Hill equation embedded in the nonlinear model. We further evaluated our estimation performance with different parameter settings and signal to noise ratios. Our results demonstrated the effectiveness of the algorithm for both linearly and nonlinearly parameterized dynamic systems.
Our proposed Bayesian algorithm successfully estimated parameters in nonlinear mathematical models for biological pathways. This method can be further extended to high order systems and thus provides a useful tool to analyze biological dynamics and extract information using temporal data.
生物实验中时间测量的可用性显著推动了系统生物学的研究领域。为了深入了解生物系统的相互作用和调控，诸如常微分方程等数学框架已被广泛应用于对生物途径进行建模并解释时间数据。希尔方程是在微分方程框架中表示反应速率的首选形式，因为它们结构简单且能够轻松拟合饱和实验测量值。然而，希尔方程是高度非线性参数化的函数，这些函数中的参数不易测量。此外，由于其高度非线性，为线性参数化微分方程开发的自适应参数估计算法无法应用。因此，生物途径的非线性参数化微分方程模型中的参数估计既具有挑战性又很有意义。在本研究中，我们提出一种贝叶斯参数估计算法，用于使用时间序列数据估计生物途径非线性数学模型中的参数。
我们使用龙格 - 库塔方法，在假设微分方程已知结构的情况下将微分方程转换为差分方程。这种转换使我们能够生成依赖于先前状态的预测，并应用贝叶斯方法，即马尔可夫链蒙特卡罗（MCMC）方法。我们将此方法应用于左心室（LV）对心肌梗死（MI）反应所涉及的生物途径，并通过估计嵌入非线性模型的希尔方程中的两个参数来验证我们的算法。我们进一步用不同的参数设置和信噪比评估了我们的估计性能。我们的结果证明了该算法对于线性和非线性参数化动态系统均有效。
我们提出的贝叶斯算法成功估计了生物途径非线性数学模型中的参数。该方法可进一步扩展到高阶系统，从而为使用时间数据分析生物动力学和提取信息提供了一个有用的工具。
