Inferring dynamics of metabolic networks directly from metabolomics data provides a promising way to elucidate the underlying mechanisms of biological systems, as reported in our previous studies [1-3] by a differential Jacobian approach. The Jacobian is solved from an over-determined system of equations as JC + CJT = -2D, called Lyapunov Equation in its generic form , where J is the Jacobian, C is the covariance matrix of metabolomics data and D is the fluctuation matrix. Lyapunov Equation can be further simplified as the linear form Ax = b. Frequently, this linear equation system is ill-conditioned, i.e., a small variation in the right side b results in a big change in the solution x, thus making the solution unstable and error-prone. At the same time, inaccurate estimation of covariance matrix and uncertainties in the fluctuation matrix bring biases to the solution x. Here, we firstly reviewed common approaches to circumvent the ill-conditioned problems, including total least squares, Tikhonov regularization and truncated singular value decomposition. Then we benchmarked these methods on several in-silico kinetic models with small to large perturbations on the covariance and fluctuation matrices. The results identified that the accuracy of the reverse Jacobian is mainly dependent on the condition number of A, the perturbation amplitude of C and the stiffness of the kinetic models. Our research contributes a systematical comparison of methods to inversely solve Jacobian from metabolomics data
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