Stationary states of genome scale metabolic networks in continuous cell cultures Roberto Mulet Group of Complex Systems and Statistical Physics, Physics Faculty. University of Havana, CUBA With the support of the RISE-H2020 project, 2017-2021
reservoir e ffl uent culture vessel
Requirements ◮ It must include the internal metabolism ◮ It must include the chemostat ◮ Be computationally scalable to Genome scale metabolic networks ◮ Be flexible ◮ Toxicity ◮ Heterogeneity
Outline Homogeneous chemostat Mathematical framework Stationary States From a Toy model to Genome Scale Heterogeneus chemostat Maximum Entropy Principle The Toy model again Genome Scale Metabolic Network Conclusions
Homogeneous chemostat Mathematical framework Stationary States From a Toy model to Genome Scale Heterogeneus chemostat Maximum Entropy Principle The Toy model again Genome Scale Metabolic Network Conclusions
Chemostat dX dt = ( µ − σ − D ) X (1) µ = µ ( ν ) σ = σ ( s ) (2) ds i dt = − u i X − ( s i − c i ) D (3)
The cell lb k ≤ r k ≤ ub k
The cell lb k ≤ r k ≤ ub k X } = min { V i , c i D − L i ≤ u i ≤ min { V i , c i ξ }
The cell lb k ≤ r k ≤ ub k X } = min { V i , c i D − L i ≤ u i ≤ min { V i , c i ξ } � r k < K k
The cell lb k ≤ r k ≤ ub k X } = min { V i , c i D − L i ≤ u i ≤ min { V i , c i ξ } � r k < K k � S ik r k − e i − y i µ + u i = 0 (4) k
The cell lb k ≤ r k ≤ ub k X } = min { V i , c i D − L i ≤ u i ≤ min { V i , c i ξ } � r k < K k � S ik r k − e i − y i µ + u i = 0 (4) k This is a polytope in very high dimensions
The cell lb k ≤ r k ≤ ub k X } = min { V i , c i D − L i ≤ u i ≤ min { V i , c i ξ } � r k < K k � S ik r k − e i − y i µ + u i = 0 (4) k The cell maximizes biomass production µ Linear Programming LP
Mathematical framework lb k ≤ r q ≤ ub k dX dt = ( µ − σ − D ) X D − L i ≤ u i ≤ min { V i , c i X } � r k < K µ = µ ( ν ) σ = σ ( s ) k � S ik r k − e i − y i µ + u i = 0 k ds i dt = − u i X − ( s i − c i ) D The cell maximizes biomass production LP
Flux Balance � S ik r k − e i − y i µ + u i = 0 k lb k ≤ r k ≤ ub k X } = min { V i , c i D − L i ≤ u i ≤ min { V i , c i ξ } � r i < K i
Flux Balance � S ik r k − e i − y i µ + u i = 0 k lb k ≤ r k ≤ ub k X } = min { V i , c i D − L i ≤ u i ≤ min { V i , c i ξ } � r i < K i The cell maximizes biomass production µ LP u ∗ i ( ξ ) . . . µ ( ξ )
Equilibrium in metabolite’s concentration ds i dt = − u ∗ i X − ( s i − c i ) D
Equilibrium in metabolite’s concentration ds i dt = − u ∗ i X − ( s i − c i ) D s ∗ i ( ξ ) = c i − u ∗ i ( ξ ) ξ
Stationarity in cell’s concentration dX dt = ( µ − σ − D ) X
Stationarity in cell’s concentration dX dt = ( µ − σ − D ) X 0 = ( µ ∗ ( ξ ) − σ ∗ ( ξ ) − D ) X ∗
Stationarity in cell’s concentration dX dt = ( µ − σ − D ) X D = µ ∗ ( ξ ) − σ ∗ ( ξ )
Stationarity in cell’s concentration dX dt = ( µ − σ − D ) X X ∗ = µ ∗ ( ξ ) − σ ∗ ( ξ ) ξ
Stationarity equations r ∗ k . . . u ∗ i ( ξ ) . . . µ ∗ ( ξ ) s ∗ i ( ξ ) = c i − u ∗ i ( ξ ) ξ X ∗ ( ξ ) = µ ∗ ( ξ ) − σ ∗ ( ξ ) ξ
Small Network S E P W E Vazquez et al.. Macromolecular crowding explains overflow metabolism in cells. Scientific Reports 6, 31007 (2016)
Toxicity is the key point bistable regime (a) (b)
General Picture S S E E P W P W E E respiration over fl ow (a) Overflow. At high enough nutrient uptake the respiratory flux hit s the upper bound r max and the remaining nutrients are exported as W . (b) Respiration. The nutrient is completely oxidized with a large energy yield. (c) Threshold values of ξ . ξ 0 delimits the nutrient excess regime ( ξ < ξ 0 ) from the competition regime ( ξ > ξ 0 ). ξ sec delimits the transition between overflow metabolism ( ξ < ξ sec and respiration ( ξ > ξ sec ). Finally, maintenance demand cannot be met beyond ξ > ξ m .
Genome Scale: CHO-K1 line ◮ 6663 reactions ◮ V glc = 0 . 5 mmol /gDW/h ◮ V i = . 1 V glc
Metabolite uptakes and concentration
General picture of the transitions limiting: limiting: gly, tyr, limiting: max. yield nutrient excess ser, asp, pro trp, his, arg, lys, phe glc, gln, asn aas aas glc aas aas aas asp asp glc glc asp glc asp glc asp succ for succ for acald acald for acald lac ala pyr lac for ala ala lac ala
Steady state and bifurcation bistable regime (a) (b) J. Fernandez-de-Cossio Diaz, K. Le´ on and R. M. , Characterizing stationary states of genome scale metabolic networks in continuous culture, PLOS Computational Biology. 13 (11): e1005835 (2017)
Homogeneous chemostat Mathematical framework Stationary States From a Toy model to Genome Scale Heterogeneus chemostat Maximum Entropy Principle The Toy model again Genome Scale Metabolic Network Conclusions
Constraints � S ik r k − e i − y i µ + u i = 0 k lb k ≤ r q ≤ ub k D − L i ≤ u i ≤ min { V i , c i X } � r i < K i We must explore this polytope
Stationarity: Dealing with the heterogeneity d � X dt = ( µ − σ − D ) � X
Stationarity: Dealing with the heterogeneity d � X dt = ( µ − σ − D ) � X 0 = ( µ ( ν ) − σ ( s ) − D )
Stationarity: Dealing with the heterogeneity d � X dt = ( µ − σ − D ) � X 0 = ( µ ( ν ) − σ ( s ) − D ) Effetive Growth rate = µ ( ν ) − σ ( s ) = D
Maximum Entropy Principle If s is fixed, µ ( ν ) − σ ( s ∗ ) = D
Maximum Entropy Principle P s ∗ ( ν ) ∼ e β ( µ ( ν ) − σ ( s ∗ ))
Maximum Entropy Principle P s ∗ ( ν ) ∼ e β ( µ ( ν ) − σ ( s ∗ )) s i = c i − 1 � u a i D a
Maximum Entropy Principle P s ∗ ( ν ) ∼ e β ( µ ( ν ) − σ ( s ∗ )) i = c i − X � s ∗ u i ( ν ) P s ∗ ( ν ) d ν D Π
In short � � � e β ( µ ( ν ) − σ ( s ∗ )) µ ( ν ) − σ ( s ∗ ) D = X Π d ν ξ = � Π d ν e β ( µ ( ν ) − σ ( s ∗ ))
In short � � � e β ( µ ( ν ) − σ ( s ∗ )) µ ( ν ) − σ ( s ∗ ) D = X Π d ν ξ = � Π d ν e β ( µ ( ν ) − σ ( s ∗ )) � s ∗ i = c i − ξ u i ( ν ) P s ∗ ( ν ) d ν Π
Homogeneous vs Heterogeneous Chemostat D = X ξ = � µ ( ν ) − σ ( s ∗ ) � P s ∗ X ∗ ( ξ ) = µ ∗ ( ξ ) − σ ∗ ( ξ ) ξ � s ∗ i = c i − ξ u i ( ν ) P s ∗ ( ν ) d ν s ∗ i ( ξ ) = c i − ξ u ∗ i ( ξ ) Π
Summarizing � S ik r k − e i − y i µ + u i = 0 k lb k ≤ r q ≤ ub k D − L i ≤ u i ≤ min { V i , c i X } D = � µ ( ν ) − σ ( s ∗ ) � P s ∗ � r i < K i = c i − X � s ∗ u i ( ν ) P s ∗ ( ν ) d ν i D Π
Small Network again S E P W E
Effect of the heterogeneity ∞
Effect of the heterogeneity X (10 6 cells/mL) unfeasible 2.0 1.5 stable 1.0 0.5 unstable 0
Genome Scale: CHO-K1 line ◮ 6663 reactions ◮ V glc = 0 . 5 mmol /gDW/h ◮ V i = . 1 V glc
Exploring the space Π � S ik r k − e i − y i µ + u i = 0 k lb k ≤ r q ≤ ub k D − L i ≤ u i ≤ min { V i , c i X } � r i < K i
Exploring the space Π � S ik r k − e i − y i µ + u i = 0 k lb k ≤ r q ≤ ub k D − L i ≤ u i ≤ min { V i , c i X } � r i < K i For β = ∞ : Expectation Propagation Alfredo Braunstein, Anna Paola Muntoni, Andrea Pagnani, An analytic approximation of the feasible space of metabolic networks, Nat. Comm. 8 , 14915 (2017) Here generalized for finite β
Genome Scale Metabolic Networks λ m β = ∞ λ m β =0 λ m β =0 a) b) c) λ m β >970 s glc (mM) s nh4 (mM) λ m β =970 s lac (mM) λ m β = ∞ β =0 λ m β =970 0.01 0.1 1 10 100 1000 0.01 0.1 1 10 100 1000 0.01 0.1 1 10 100 1000 ξ (10 6 cells day/mL) ξ (10 6 cells day/mL) ξ (10 6 cells day/mL)
Genome Scale Metabolic Networks 1.5 λ m β =970 X (10 6 cells/mL) 1.0 λ m β = ∞ 0.5 λ m β =97 0.0 0.5 1.0 1.5 2.0 D (1/day) J. Fernandez-de-Cossio Diaz, and R. M. , Maximum Entropy and Population Heterogeneity in continuos cell cultures , arXiv:1807.04218
Homogeneous chemostat Mathematical framework Stationary States From a Toy model to Genome Scale Heterogeneus chemostat Maximum Entropy Principle The Toy model again Genome Scale Metabolic Network Conclusions
Conclusions ◮ We developed a mathematical framework to determine the stationary states in a chemostat ◮ The presence of toxic waste: ◮ drives the appareance of many stationary states ◮ makes relevant the history of the system ◮ We provided a scheme to estimate the metabolic flux distribution of an heterogeneous culture in a chemostat ◮ The presence of heterogeneity in the culture ◮ changes the concentration of metabolites ◮ allows stationary states with a larger number of cells ◮ Everything is computationally tractable in Genome Scale metabolic networks
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