Abstract
Constraint-Based Reconstruction and Analysis (COBRA) is currently the only methodology that permits integrated modeling of Metabolism and macromolecular Expression (ME) at genome-scale. Linear optimization computes steady-state flux solutions to ME models, but flux values are spread over many orders of magnitude. Data values also have greatly varying magnitudes. Standard double-precision solvers may return inaccurate solutions or report that no solution exists. Exact simplex solvers based on rational arithmetic require a near-optimal warm start to be practical on large problems (current ME models have 70,000 constraints and variables and will grow larger). We have developed a quadruple-precision version of our linear and nonlinear optimizer MINOS, and a solution procedure (DQQ) involving Double and Quad MINOS that achieves reliability and efficiency for ME models and other challenging problems tested here. DQQ will enable extensive use of large linear and nonlinear models in systems biology and other applications involving multiscale data.
Highlights
Metabolism and macromolecular Expression (ME) models have opened a whole new vista for predictive mechanistic modeling of cellular processes, but their size and multiscale nature pose a challenge to standard linear optimization (LO) solvers based on 16-digit double-precision floating-point arithmetic
We developed a Double-Quad-Quad MINOS procedure (DQQ) that combines the use of Double and Quad solvers in order to achieve a balance between efficiency in computation and accuracy of the solution
For M models, we find that Double MINOS alone is sufficient to obtain non-zero steady-state solutions that satisfy feasiblility and optimality conditions with a tolerance of 10−7
Summary
Metabolism and macromolecular Expression (ME) models have opened a whole new vista for predictive mechanistic modeling of cellular processes, but their size and multiscale nature pose a challenge to standard linear optimization (LO) solvers based on 16-digit double-precision floating-point arithmetic. Lifting reduces the largest matrix entries by introducing auxiliary constraints and variables This approach has permitted standard (double-precision) LO solvers to find more accurate solutions, even though the final objective value is still not satisfactory. (For Single variables a and b, Fortran compilers would use Single arithmetic to evaluate the basic expressions a ±b, a*b, a/b, whereas C compilers would transfer a and b to longer registers and operate on them using Double arithmetic.) Most often, the C compiler’s extra precision was not needed, but occasionally it did make a critical difference Kahan calls this the humane approach to debugging complex numerical software. We believe the time has come to produce Quad versions of key sparse-matrix packages and large-scale optimization solvers for multiscale problems
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