Processes 2021, 9, 1–9; doi:https://doi.org/10.3390/pr9020322 S1 of S9 Supplementary Materials: Supplementary Materials for “Approaches to Model Cell Metabolism: A Review on Constraint-based, and Kinetic Metabolic Modeling” Mohammadreza Yasemi , Mario Jolicoeur * Text S1: Mathematical formulations of constraint-based models Stoichiometric matrix formulation The stoichiometric matrix for a metabolic network with m metabolites and n reactions is as follows: Sm×n = si,j fi = 1, ..., mjj = 1, ..., ng (1) m×n n n dxi + − = ∑ sij vj − ∑ sij vj, i = 1, . , m (2) dt j=1 j=1 Therefore, we have one column vector for each reaction and Eq.2 can be written in matrix-vector notation for the whole metabolic network as follows: d~x = S × ~v (3) dt m n The null space of stoichiometric matrix formulation Where r is the rank for stoichiometric matrix S, the basis for the null space is a set with q = (n − r) linearly independent column vectors of dimension n, which generates the null space for stoichiometric matrix S when spanned [1]. ~ n ~ ~ ~ ~ ~ K (S) = fbi 2 R (i = 1, . , q) j S.bi = 0 and bi.bj = 0 (i 6= j)g (4) ~ ~ Null space(S) = spanfb1,..., bqg, q = (n − r) (5) Metabolic Flux Analysis (MFA) MFA provides an empirical flux map: S.~v = 0 " # h i ~vu Su Sk = 0 (6) ~vk T −1 T ~vu = −[(Su Su) Su ]Sk~vk Processes 2021, 9 ; doi:https://doi.org/10.3390/pr9020322 S2 of S9 Flux Balance Analysis (FBA) The mathematical formulation of a cell objective, accounting for defined fluxes restrictions with bounding limits for the reaction fluxes, is developed as a linear programming optimization problem for FBA: max C~T.~v subject to: (7) S.~v = 0 lb < ~v < ub Modeling approaches complying to the thermodynamics-based constraints formulation The formulation of TMFA combines constraints from FBA, directionality and constraints on metabolites concentrations as follows: S.~v = 0, (8) 0 ≤ vi ≤ zivMax, fi = 1, ..., rg, (9) 0 0◦ DrGi − K + Kzi < 0, fi = 1, ..., rjDrGi is knowng, (10) m 0◦ 0 0◦ DrGi + RT ∑ ni,j ln xj = DrGi , fi = 1, ..., r + LjDrGi is knowng, (11) j=1 0 DrGi − Kyi < 0, fi = 1, ..., r + Lg, (12) r r yi + ∑ ai,jzj ≤ ∑ ai,j, fi = 1, ..., r + Lg (13) j=1 j=1 In this method, each reversible reaction is decomposed to two backward and forward reactions. Therefore, there will be no negative vi value. zi is a binary number equal to one when there is non-zero flux or otherwise equals to zero. In Eq 10, K is fixed as a value to make sure the inequality is satisfied when zi and vi are zeros. Of interest, the whole inequality checks that the solution for flux distribution obeys the second law of thermodynamics. The Gibbs free energy calculation for each reaction i is carried out in Eq 11 with considering the activity xj of involved metabolites. Processes 2021, 9 ; doi:https://doi.org/10.3390/pr9020322 S3 of S9 Figure S1: EFMs enumeration for the running example with reversible reactions S1 EFMs enumeration for the running example.pdf Figure S1. EFMs enumeration for the running example with reversible reactions (a) The balanced metabolic network is same as in the main text. (b) The basis for the null space is the kernel matrix of the stoichiometric matrix. (c) 7 EFMs are enumerated assuming that all the reactions are reversible, except for v6 which is inward. Each EFM is a linear combination of the basis vectors and for this system with reversible reactions, the negative coefficients in the columns 6 and 7 appear. (d-j) The flux maps for the EFMs are shown on the substrate graph of the network. (g,i,j) The EFMs that include some reactions in the reverse direction. (h) The EFM that includes a futile cycle. Processes 2021, 9 ; doi:https://doi.org/10.3390/pr9020322 S4 of S9 Table S1. Properties of the different MPA methods. MPA Methods Properties Method Advantages Disadvantages Applications Toolbox/Software Name Elementary No need for flux measurement Calculation of all EFMs is Suitable for small to medium scale models [4, METATOOL [13]; Flux Mode and optimality objective function computationally demanding 6]; Identification of coregulated and coexpressed CellNetAnalyzer (EFM) [2]; Every flux distribution can be [4,5]; Small percentage of EFMs are reactions and genes [7]; Composition of the [14]; COPASI decomposed into basic functional biologically relevant [4] minimal substrate required [7–9]; Quantification [15]; Efmtool[16]; units without cancellation [3,4] of cellular robustness [4,10,11]; Study of gene YANAsquare [17] deletions and knockouts [4,12]; Finding operational modes and optimal routes [12]; Finding pathway length [12] Extreme No need for flux measurement and Fluxes can cancel out [4,18,19]; Do Suitable for small to medium scale models [4,6]; To Expa [21]; Pathway optimality objective function [2]; not decompose reversible exchange caluclate the edges of solution space cone [18] METATOOL [13]; Analysis Reducing number of pathways and reactions [20]; Not suitable to study CellNetAnalyzer [14]; (EPA) applicability for assessing network mutations and the effect of reaction YANAsquare [17] properties [12] removal [12] Minimal cut Unbiased solution space with MCS may eliminate desired To predict metabolic reaction candidates for OptKnock [25]; set (MCS)/ respect to optimality objectives; functionalities of a metabolic elimination in order to couple cell growth and CellNetAnalyzer constrained Exploits minimum desired yields of network mistakenly [24]; The production [23]; The applicability is extended to (version 2018.2) [14] minimal cut bioproducts and minimum desired computation cost is still higher than genome-scale models [22] set (cMCS) biomass yield as additional inputs to FBA-based methods constrain the design space; The flux distributions are obligated to only represent growth that is coupled with production [22,23] Table S2. The hypothesized objective functions of the cell and their biological rationales. Particular Objectives of the Cell Objective Biological rationale Reference Maximum growth In the exponential growth phase, cells utilize their [26] resources to maximize proliferation, i.e. to form new biomass. This assumption is biologically relevant for both prokaryotes and eukaryotes, although a large part of cell resources are directed to maintenance cell in eukaryotes. Maximum bioenergetic Cells aim for the maximal ATP production, or [27,28] production maximum reducing power, which dictates a distinct distribution of carbon flux. Minimum overall Cells aim for the minimum sum of the squares of [29–31] intracellular flux fluxes, based on the assumed maximum enzymatic efficiency for cellular growth. Maximum product Recombinant DNA (rDNA) facilitates the design [32] formation of modified cell lines with desired traits beyond predetermined cell objectives. Minimum reaction steps Assuming that the cell chooses the shortest [33] metabolic path from a substrate to a product. Minimum redox potential Focus cell capability to generate energy through [34] ubiquitous redox reactions. Processes 2021, 9 ; doi:https://doi.org/10.3390/pr9020322 S5 of S9 Table S3. List of the flux balance analysis enhancements. Enhancements of Flux Balance Analysis Method Name Approach Regulation Enzyme-cost Reference application Flux Balance Analysis (FBA) Steady-state NO NO Studying gene knockouts [35,36], Studying cell constraint-based growth on different media [35], Finding metabolic gaps [37] Dynamic FBA (dFBA) Iterative or NO NO Description of unsteady-state growth and dynamic by-product secretion in aerobic batch, fed-batch, and anaerobic batch cultures [38], analysis of diauxic growth in Escherichia coli [38], study of carbon storage metabolism in microalgae [39] Dynamic enzyme-cost FBA Dynamic NO YES Prediction of dynamic changes in metabolic (deFBA) fluxes and biomass composition during metabolic adaptations [40] Resource balance analysis Steady-state NO YES Prediction of the cell composition of bacteria with (RBA) respect to their medium [41] Dynamic resource balance Dynamic NO YES Optimization of production of added-value analysis (dRBA) compounds by bacteria [42] Regulatory FBA (rFBA) Iterative, YES NO Regulation of gene expression [43,44], predict Boolean logic of high-throughput experiments outcome [45], indication of knowledge gaps and unknown components and interactions [45], identification of potential targets for transcription factors [46] Steady-state regulatory FBA Steady-state YES NO Prediction of gene expression and metabolic fluxes (SR-FBA) [47] Probabilistic regulation of Steady-state, YES NO Statistical inference of regulatory network, metabolism (PROM) probabilistic automated quantification of regulatory Interactions [48] Integrated FBA (iFBA) Iterative, YES NO Encapsulation of dynamics of internal metabolites, Boolean logic, consideration of signaling molecules [49] ODE Integrated dynamic FBA Iterative, YES YES Integration of metabolic, signaling and regulatory (idFBA) Boolean logic, networks [50] ODE Parsimonious FBA (pFBA) Steady-state NO YES describing metabolic states of the cell by minimizing the overall enzymatic fluxes to eliminate the alternative flux solutions [51] Enzyme-cost FBA (ecFBA) Steady-state YES YES incorporating enzyme kinetic information in FBA framework, reduces the flux variability [52] Regulatory dynamic Dynamic YES YES Consideration of enzyme production cost and enzyme-cost FBA (r-deFBA) regulatory events, integration of metabolic, signaling and regulatory networks [53] Table S4. Comparison of MFA and FBA. Metabolic Flux Analysis (MFA) Flux Balance Analysis (FBA) −1 T ~vu = −Su Sm~vm max~c .~v; subject to: S.~v = 0 and lb < ~v < ub ~vu: unknown flux vector; ~v: flux vector; ~vm: measured flux vector; ~c: weights associated to fluxes based on the Su , Sm: stoichiometric matrix subsets for unknown and objective of optimization; measured fluxes respectively [54]. lb , ub: allowed lower and upper bound respectively for the flux values[35,55]. Applicable for small to medium scale metabolic networks Applicable for small to genome-scale metabolic networks at steady-state. For calculation of the empirical value of at steady-state.