Stoich of cell growth

Overall balance for cells growing on carbohydrate

CHmOn + a O2 + b NH3 → c CHαOβNδ + d H2O + e CO2

Components of overall cell-growth balance

Carbon source/substrate; oxygen; nitrogen source; biomass/cells; water; carbon dioxide

Carbon dioxide production during metabolism

CO2 is made at the end of glycolysis/beginning of the TCA cycle when pyruvate is converted to Acetyl-CoA and also during the TCA cycle

CO2 is not produced at

The end of the respiratory chain when electrons are transferred to O2

Mass balance for cell growth with product

CHmOn + a O2 + b NH3 → c CHαOβNδ + d CHxOyNz + e H2O + f CO2

Components of cell growth with product balance

Carbon source/substrate; nitrogen source; biomass/cells; product

Coefficient convention

The coefficient for the carbon source can be set to 1 when its formula contains one carbon; the biomass coefficient can also be set to 1 depending on how the equation is written

Elemental balance: Carbon

1 = c + d + f

Elemental balance: Hydrogen

m + 3b = cα + dx + 2e

Elemental balance: Oxygen

n + 2a = cβ + dy + e + 2f

Elemental balance: Nitrogen

b = cδ + dz

Empirical chemical formula for cell

Defined per gram atom (mole) of carbon as CHαOβNδ

Typical biomass empirical formula values

α ≈ 1.64–2; β ≈ 0.27–0.56; δ ≈ 0.13–0.25

Unknown stoichiometric coefficients

a

Known information needed

Chemical formulas for the carbon source

Mass balance when carbon source and product contain more than one carbon

CHmOn + a O2 + b NH3 → c CHαOβNδ + d CwHxOyNz + e H2O + f CO2

Carbon balance when carbon source has more than one carbon

k = c + dw + f

Hydrogen balance when carbon source has more than one carbon

m + 3b = cα + dx + 2e

Oxygen balance when carbon source has more than one carbon

n + 2a = cβ + dy + e + 2f

Nitrogen balance when carbon source has more than one carbon

b = cδ + dz

Problem with elemental balances

There are six unknowns a

Alternative equation convention

Sometimes the equation is written so the formulas for the carbon source and product contain more than one carbon atom

Alternative balance form

a CkHmOn + b O2 + c NH3 → CHαOβNδ + d CwHxOyNz + e H2O + f CO2

Carbon balance in alternative form

ka = 1 + dw + f

Hydrogen balance in alternative form

am + 3c = α + dx + 2e

Oxygen balance in alternative form

an + 2b = β + dy + e + 2f

Nitrogen balance in alternative form

c = δ + dz

Respiratory quotient (RQ)

A measurable quantity defined as the molar rate of CO2 production divided by the molar rate of O2 consumption

RQ equation

RQ = mol CO2 production / mol O2 consumption

RQ and stoichiometric coefficients

RQ is proportional to the molar coefficients on O2 and CO2 in the overall balance equation

RQ relationship

f = RQ × a

Why RQ is useful

RQ provides a fifth equation along with the carbon

Overall balance with RQ

CHmOn + a O2 + b NH3 → c CHαOβNδ + d CHxOyNz + e H2O + f CO2

Growth yield YX/S

Yield coefficient for cell growth per substrate; grams of biomass produced per gram of substrate consumed

YX/S equation

YX/S = ΔX / -ΔS

Why YX/S is not 1

Not all carbon in the carbon source is converted to cell biomass; a fraction is respired as CO2 during transformation of carbon to energy (ATP)

Substrate energy distribution

Energy from substrate is divided among biomass formation

Typical aerobic YX/S

0.4 to 0.6 g cells/g glucose

Anaerobic YX/S

Lower than the typical aerobic value

Yield coefficient based on oxygen

YX/O2 is the yield coefficient for cell growth based on oxygen

Typical YX/O2

0.9 to 1.4 g cells/g O2 for most yeast and bacteria

Reduced substrates and YX/O2

YX/O2 is lower for highly reduced substrates such as methane

YX/O2 equation

YX/O2 = ΔX / -ΔO2

Meaning of YX/O2

Change in mass of biomass (X) per change in mass of O2

Oxygen sign convention

O2 is consumed

Substrate sign convention

S is consumed

Product yield based on substrate

YP/S is the yield coefficient for product based on substrate

YP/S equation

YP/S = ΔP / -ΔS

Meaning of YP/S

Change in mass of product per change in mass of substrate

Product yield sign convention

S is consumed

Worked microorganism example

A microorganism with formula C18H29O3N3.6 and molecular weight 343.4 g/mol is grown in a batch reactor on a pentose sugar C5H10O5 with molecular weight 150 g/mol for production of ethanol C2H6O with molecular weight 46.0 g/mol

Worked mass balance equation

C5H10O5 + a O2 + b NH3 → c C18H29O3N3.6 + d C2H6O + e H2O + f CO2

Worked carbon balance

5 = c(18) + d(2) + f

Worked hydrogen balance

10 + 3b = c(29) + d(6) + e(2)

Worked oxygen balance

5 + a(2) = c(3) + d + e + f(2)

Worked nitrogen balance

b = c(3.6)

Problem with worked example

There are not enough equations to solve for a

Worked-example yield data

YX/S = 0.229 g cells/g S; YP/S = 0.2147 g P/g S

Meaning of YX/S in worked example

0.229 g cells are produced for each gram of substrate consumed

Meaning of YP/S in worked example

0.2147 g product is produced for each gram of substrate consumed

Calculating moles of cells

Use YX/S and the molecular weights of cells and substrate to convert the yield into moles of cells produced (stoichiometric coefficient c) for each mole of substrate consumed

Calculating moles of product

Use YP/S to calculate the moles of product made (stoichiometric coefficient d) for each mole of substrate consumed

Worked cell coefficient calculation

0.229 g cells/g S × 150 g S/mol S × 1 mol cells/343.4 g cells = approximately 0.1 mol cells/mol S

Worked product coefficient calculation

0.2147 g product/g S × 150 g S/mol S × 1 mol product/46 g product = approximately 0.7 mol product/mol S

Class stopping point

The class stopped here on Friday; the problem will be finished during the next class