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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 culture vessel effluent

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) dsi dt = −uiX − (si − ci)D (3)

The cell

lbk ≤ rk ≤ ubk

The cell

lbk ≤ rk ≤ ubk −Li ≤ ui ≤ min{Vi, ci D X } = min{Vi, ci ξ }

The cell

lbk ≤ rk ≤ ubk −Li ≤ ui ≤ min{Vi, ci D X } = min{Vi, ci ξ }

• k

rk < K

The cell

lbk ≤ rk ≤ ubk −Li ≤ ui ≤ min{Vi, ci D X } = min{Vi, ci ξ }

• k

rk < K

• k

Sikrk − ei − yiµ + ui = 0 (4)

The cell

lbk ≤ rk ≤ ubk −Li ≤ ui ≤ min{Vi, ci D X } = min{Vi, ci ξ }

• k

rk < K

• k

Sikrk − ei − yiµ + ui = 0 (4) This is a polytope in very high dimensions

The cell

lbk ≤ rk ≤ ubk −Li ≤ ui ≤ min{Vi, ci D X } = min{Vi, ci ξ }

• k

rk < K

• k

Sikrk − ei − yiµ + ui = 0 (4) The cell maximizes biomass production µ Linear Programming LP

Mathematical framework

dX dt = (µ − σ − D)X µ = µ(ν) σ = σ(s) dsi dt = −uiX − (si − ci)D lbk ≤ rq ≤ ubk −Li ≤ ui ≤ min{Vi, ci D X }

• k

rk < K

• k

Sikrk − ei − yiµ + ui = 0 The cell maximizes biomass production LP

Flux Balance

• k

Sikrk − ei − yiµ + ui = 0 lbk ≤ rk ≤ ubk −Li ≤ ui ≤ min{Vi, ci D X } = min{Vi, ci ξ }

• i

ri < K

Flux Balance

• k

Sikrk − ei − yiµ + ui = 0 lbk ≤ rk ≤ ubk −Li ≤ ui ≤ min{Vi, ci D X } = min{Vi, ci ξ }

• i

ri < K The cell maximizes biomass production µ LP u∗

i (ξ) . . . µ(ξ)

Equilibrium in metabolite’s concentration

dsi dt = −u∗

i X − (si − ci)D

Equilibrium in metabolite’s concentration

dsi dt = −u∗

i X − (si − ci)D

s∗

i (ξ) = ci − 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 ∗ ξ = µ∗(ξ) − σ∗(ξ)

Stationarity equations r ∗

k . . . u∗ i (ξ) . . . µ∗(ξ)

s∗

i (ξ) = ci − u∗ i (ξ)ξ

X ∗(ξ) ξ = µ∗(ξ) − σ∗(ξ)

Small Network

S E P E W

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 E P E W respiration S E P E W

• verflow

(a) Overflow. At high enough nutrient uptake the respiratory flux hit s the upper bound rmax 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. ξ0 delimits the nutrient excess regime (ξ < ξ0) from the competition regime (ξ > ξ0). ξsec delimits the transition between overflow metabolism (ξ < ξsec and respiration (ξ > ξsec). Finally, maintenance demand cannot be met beyond ξ > ξm.

Genome Scale: CHO-K1 line

◮ 6663 reactions ◮ Vglc = 0.5mmol/gDW/h ◮ Vi = .1Vglc

Metabolite uptakes and concentration

General picture of the transitions

aas glc for ala pyr succ asp aas glc for ala lac succ asp aas glc ala acald asp lac aas glc ala acald asp lac for limiting: ser, asp, pro nutrient excess limiting: gly, tyr, trp, his, arg, lys, phe limiting: glc, gln, asn aas glc acald asp for

• max. yield

Steady state and bifurcation

(a) (b)

bistable regime

• J. Fernandez-de-Cossio Diaz, K. Le´
• n 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

• k

Sikrk − ei − yiµ + ui = 0 lbk ≤ rq ≤ ubk −Li ≤ ui ≤ min{Vi, ci D X }

• i

ri < K 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

Ps∗(ν) ∼ eβ(µ(ν)−σ(s∗))

Maximum Entropy Principle

Ps∗(ν) ∼ eβ(µ(ν)−σ(s∗)) si = ci − 1 D

• a

ua

i

Maximum Entropy Principle

Ps∗(ν) ∼ eβ(µ(ν)−σ(s∗)) s∗

i = ci − X

D

• Π

ui(ν)Ps∗(ν)dν

In short

D = X ξ =

• Π dν
• µ(ν) − σ(s∗)
• eβ(µ(ν)−σ(s∗))
• Π dνeβ(µ(ν)−σ(s∗))

In short

D = X ξ =

• Π dν
• µ(ν) − σ(s∗)
• eβ(µ(ν)−σ(s∗))
• Π dνeβ(µ(ν)−σ(s∗))

s∗

i = ci − ξ

• Π

ui(ν)Ps∗(ν)dν

Homogeneous vs Heterogeneous Chemostat

D = X ξ = µ(ν) − σ(s∗)Ps∗ s∗

i = ci − ξ

• Π

ui(ν)Ps∗(ν)dν X ∗(ξ) ξ = µ∗(ξ) − σ∗(ξ) s∗

i (ξ) = ci − ξu∗ i (ξ)

Summarizing

D = µ(ν) − σ(s∗)Ps∗ s∗

i = ci − X

D

• Π

ui(ν)Ps∗(ν)dν

• k

Sikrk − ei − yiµ + ui = 0 lbk ≤ rq ≤ ubk −Li ≤ ui ≤ min{Vi, ci D X }

• i

ri < K

Small Network again

S E P E W

Effect of the heterogeneity

∞

Effect of the heterogeneity

X (106 cells/mL)

unstable stable unfeasible

0.5 1.0 1.5 2.0

Genome Scale: CHO-K1 line

◮ 6663 reactions ◮ Vglc = 0.5mmol/gDW/h ◮ Vi = .1Vglc

Exploring the space

Π

• k

Sikrk − ei − yiµ + ui = 0 lbk ≤ rq ≤ ubk −Li ≤ ui ≤ min{Vi, ci D X }

• i

ri < K

Exploring the space

Π

• k

Sikrk − ei − yiµ + ui = 0 lbk ≤ rq ≤ ubk −Li ≤ ui ≤ min{Vi, ci D X }

• i

ri < K

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

sglc (mM)

ξ (106 cells day/mL)

λmβ=0 λmβ>970 λmβ=0 λmβ=970 β=0 λmβ=970 a) b) c) λmβ=∞ λmβ=∞

0.01 0.1 1 10 100 1000 0.01 0.1 1 10 100 1000 0.01 0.1 1 10 100 1000

snh4 (mM) slac (mM)

ξ (106 cells day/mL) ξ (106 cells day/mL)

Genome Scale Metabolic Networks

D (1/day) X (106 cells/mL)

λmβ=97

2.0 1.0 1.5 0.0 0.5 1.5 1.0 0.5

λmβ=970 λmβ=∞ 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

Collaborators and acknowledgments

◮ Jorge Fern´

andez de Cossio. Centre for Molecular Immunology-CIM and Physics Faculty, UH. Cuba

◮ Kalet Le´

• n. Centre for Molecular Immunology-CIM. Cuba

Collaborators and acknowledgments

◮ Jorge Fern´

andez de Cossio. Centre for Molecular Immunology-CIM and Physics Faculty, UH. Cuba

◮ Kalet Le´

• n. Centre for Molecular Immunology-CIM. Cuba

◮ A. Pagnani and Alfredo Braunstein. Politecnico di Torino,

• Turin. Italy

◮ Andrea de Martino. CNR-NANOTEC in Rome, and IIGM in

• Turin. Italy

◮ Daniele de Martino. IST, Viena. Austria ◮ Ernesto Chico. Centre for Molecular Immunology. Cuba