A method for tackling primal multiplicity of solutions of dynamic flux balance models

Abstract A method is presented to tackle primal multiplicity of Dynamic flux balance analysis (DFBA) which is a Linear Programming (LP) based modeling approach that assumes that the cell distributes fluxes such as to maximize a specific biological objective. When the LP problem has multiple optima, the LP solvers usually only report the first optimum that it is reached which may not fit well the experimental data. To tackle this primal multiplicity problem, the weighted primal-dual method with auxiliary parameters is used to calculate a unique time trajectory for a given set of initial conditions. Through tuning of these auxiliary parameters, a unique optimal solution can be obtained and calibrated to fit available experimental data. Beyond its capability to tackle multiplicity, the algorithm is shown to significantly improve the prediction of some metabolites in a case study of the fed-batch fermentation of Bordetella pertussis.

[1]  Richard D Braatz,et al.  A systematic approach for finding the objective function and active constraints for dynamic flux balance analysis , 2018, Bioprocess and Biosystems Engineering.

[2]  Colin Neil Jones,et al.  Lexicographic perturbation for multiparametric linear programming with applications to control , 2007, Autom..

[3]  Paul I. Barton,et al.  DFBAlab: a fast and reliable MATLAB code for dynamic flux balance analysis , 2014, BMC Bioinformatics.

[4]  Qi Zhao,et al.  Learning cellular objectives from fluxes by inverse optimization , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[5]  Paul I. Barton,et al.  An Improved Multi-parametric Programming Algorithm for Flux Balance Analysis of Metabolic Networks , 2018, Journal of Optimization Theory and Applications.

[6]  Hanif D. Sherali,et al.  Equivalent weights for lexicographic multi-objective programs: Characterizations and computations , 1982 .

[7]  Michael Shub,et al.  Newton Flow and interior Point Methods in Linear Programming , 2005, Int. J. Bifurc. Chaos.

[8]  N. Megiddo.  Pathways to the optimal set in linear programming , 1989 .

[9]  Mari Nakamura,et al.  Growth Phase- and Nutrient Limitation-Associated Transcript Abundance Regulation in Bordetella pertussis , 2006, Infection and Immunity.

[10]  F. Doyle,et al.  Dynamic flux balance analysis of diauxic growth in Escherichia coli. , 2002, Biophysical journal.

[11]  Jingkui Chen,et al.  Analyzing Clonal Variation of Monoclonal Antibody-Producing CHO Cell Lines Using an In Silico Metabolomic Platform , 2014, PloS one.

[12]  Renato D. C. Monteiro,et al.  Limiting behavior of the affine scaling continuous trajectories for linear programming problems , 1991, Math. Program..

[13]  Ali Nikdel,et al.  Identification of active constraints in dynamic flux balance analysis , 2017, Biotechnology progress.

[14]  Jeffrey D Orth,et al.  What is flux balance analysis? , 2010, Nature Biotechnology.

[15]  Osman Güler,et al.  Limiting behavior of weighted central paths in linear programming , 1994, Math. Program..

[16]  Walid Al-Gherwi,et al.  A dynamic metabolic flux balance based model of fed‐batch fermentation of bordetella pertussis , 2013, Biotechnology progress.

[17]  Jasdeep S. Mandur,et al.  Robust batch‐to‐batch optimization in the presence of model‐plant mismatch and input uncertainty , 2017 .

[18]  Ehsan Motamedian,et al.  A new algorithm to find all alternate optimal flux distributions of a metabolic network , 2015, Comput. Chem. Eng..

[19]  Shinji Mizuno,et al.  A primal—dual infeasible-interior-point algorithm for linear programming , 1993, Math. Program..

[20]  P. I. Barton,et al.  From sugars to biodiesel using microalgae and yeast , 2015 .

[21]  Hanif D. Sherali,et al.  Linear Programming and Network Flows , 1977 .

[22]  U. Sauer,et al.  Multidimensional Optimality of Microbial Metabolism , 2012, Science.

[23]  Ehsan Motamedian,et al.  LAMOS: A linear algorithm to identify the origin of multiple optimal flux distributions in metabolic networks , 2018, Comput. Chem. Eng..

[24]  Onur Seref,et al.  Enumeration and Cartesian Product Decomposition of Alternate Optimal Fluxes in Cellular Metabolism , 2017, INFORMS J. Comput..

[25]  Evangelos Simeonidis,et al.  Flux balance analysis: a geometric perspective. , 2009, Journal of theoretical biology.

[26]  R. Mahadevan,et al.  The effects of alternate optimal solutions in constraint-based genome-scale metabolic models. , 2003, Metabolic engineering.

[27]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[28]  Zsolt Darvay,et al.  A WEIGHTED-PATH-FOLLOWING METHOD FOR LINEAR OPTIMIZATION , 2002 .

[29]  Paul I. Barton,et al.  Efficient solution of ordinary differential equations with a parametric lexicographic linear program embedded , 2015, Numerische Mathematik.

[30]  Wolfram Liebermeister,et al.  The Protein Cost of Metabolic Fluxes: Prediction from Enzymatic Rate Laws and Cost Minimization , 2016, PLoS Comput. Biol..

[31]  Anthony V. Fiacco,et al.  Nonlinear programming;: Sequential unconstrained minimization techniques , 1968 .

[32]  Kieran Smallbone,et al.  Capturing the essence of a metabolic network: a flux balance analysis approach. , 2009, Journal of theoretical biology.

[33]  P I Barton,et al.  A reliable simulator for dynamic flux balance analysis , 2013, Biotechnology and bioengineering.

[34]  H. Holzhütter.  The principle of flux minimization and its application to estimate stationary fluxes in metabolic networks. , 2004, European journal of biochemistry.

[35]  B. Palsson,et al.  Metabolic Flux Balancing: Basic Concepts, Scientific and Practical Use , 1994, Bio/Technology.

[36]  O. Mangasarian,et al.  NONLINEAR PERTURBATION OF LINEAR PROGRAMS , 1979 .

[37]  I. Grossmann,et al.  Recursive MILP model for finding all the alternate optima in LP models for metabolic networks , 2000 .

[38]  B. Palsson,et al.  Stoichiometric flux balance models quantitatively predict growth and metabolic by-product secretion in wild-type Escherichia coli W3110 , 1994, Applied and environmental microbiology.

[39]  A. Elkamel,et al.  Identification of Dynamic Metabolic Flux Balance Models Based on Parametric Sensitivity Analysis , 2017 .