Summary lectures + reader Basic Cell Factory Design
Lecture 1 - Stoichiometry
2 types of models for cells:
1. Black box model: model in which a cell is approached as a black box, so the metabolic reactions
occurring inside the cell are unknown and only the result of this reactions is known.
2. Metabolic pathway model: model in which all metabolic reactions occurring inside the cell are
known.
Past -> cell was discovered, intensively studied and then it was tried to apply the cell in a process
Nowadays -> work from process to cell, so there is a process and we look for (possibly adapted) cells
to fit in this process
Reaction which occurs inside cell is described by stoichiometric reaction equation:
rS CH2O + rN nitrogen source + rO electron acceptor + rX cells/biomass + rC CO2 + rW H2O = 0
Remarks for the stoichiometric reaction equation:
1. Reaction equation is written as a balance, so without
2. r:
r > 0 -> r = production rate CH2O = sugar (in Cmoles)
r < 0 -> r = consumption rate
So for reactants r is negative and for products r is positive
3. For simplicity no ions but only neutral compounds are included
4. Cmoles are used for all C-containing compounds, normal moles are used for compounds without
C-atoms -> so molecular formula of all C-containing compounds is written in a way that it contains 1
C-atom
5. Usually:
Nitrogen source = NH3 (ammonia)
Electron acceptor = O2 (oxygen)
Cells/biomass = CH2O0.5N0.2 (typical cell composition)
2 types of rates:
1. Volumetric rate (ri) -> [ri] = Cmolei m-3 s-1
2. Specific rate (qi) -> [qi] = Cmolei Cmolex-1 s-1
- For products -> + qi
- For reactants -> - qi
- For cells -> qx = μ (specific growth rate) -> rX = μcX
cX = concentration of cells/biomass
Finding reaction stoichiometry/all reaction rates:
1. Atom balances
2. Electron balance = shortcut to find consumption rate of electron acceptor. -> not necessarily used,
because consumption rate of electron acceptor can also be found using the atom balances
3. Pirt’s law (if you don’t have enough equations and thus need an extra equation)
,Atom balances: balances for all types of atoms which occur in the reaction equation.
Sugar (CH2O) -> used in cells for two purposes at the same time (parallel reactions):
1. Cell growth/making new cells -> by using sugar as carbon source
2. Maintain cells -> by using sugar as energy source (ATP produced using the sugar, ATP is used for
maintenance of cells)
Pirt’s law: law which describes consumption rate of sugar (rS) by cells by telling for what sugar is used
by the cells.
YXS = theoretical yield
mS = maintenance coefficient } -> determined with measurements or estimated using metabolic model
cX = concentration of cells -> can be chosen when designing process:
Higher cX? -> easier and thus cheaper DSP (downstream processing), but more sugar + electron
acceptor needed
Lower cX? -> less easy and thus less cheap DSP (downstream processing), but less sugar + electron
acceptor needed
2 types of yield which should be distinguished from each other:
1. Theoretical yield (YXS): yield which can be calculated from Pirt’s law (for Y XS) or Luedeking-Piret’s
law (for YPX), but which will never be measured in reality.
2. Observed yield (YXSOBS): yield which is actually measured in reality.
Observed yield -> defined as ratio of 2 production rates:
YXSOBS = rX/-rS (minus sign is necessary to make sure that YXSOBS is a positive number) = qX/-qS = μ/-qS
YPXOBS = rP/rX = qP/qX = qP/μ
Theoretical yield (YXS) is always bigger than the observed yield (YXSOBS), because in reality sugar is
besides for production of biomass also used for maintenance of the cells
Theoretical yield (YPX) is always smaller than the observed yield (YPXOBS), because in reality there is
also maintenance of the cells and not only biomass production -> products are also produced during
maintenance and not only during biomass production
Electron balance: balance in which the electrons going into the reaction are compared to the
electrons coming out of the reaction. -> using electron balance is useful as shortcut to find
consumption rate of electron acceptor
,Reaction equation:
Corresponding electron balance:
Electron balance can also be made
using q (specific rate) instead of r -> ri
γN, γC and γW are zero, so the electron balance becomes: in electron balance to the left is
replaced by corresponding qi
γi = degree of reduction of component i: number of electrons which is
available in component i when compared to its most oxidized form (e.g. CO2 for carbon components).
Explanation of terms:
4 * #C -> oxidation state of C in CO2 (compound in which C is most oxidized) is +4, so 4 electrons are
available by every carbon atom when γ is defined relative to an oxidation state of 0 of carbon as
reference
1 * #H -> oxidation state of H in H+ (compound in which H is most oxidized) is +1, so 1 electron is
available by every hydrogen atom when γ is defined relative to an oxidation state of 0 of hydrogen as
reference
-2 * #O -> oxidation state of O in H2O is -2, so -2 electrons are available by every oxygen atom when γ
is defined relative to an oxidation state of 0 of oxygen as reference
-3 * #N -> oxidation state of N in NH3 is -3, so -3 electrons are available by every nitrogen atom when
γ is defined relative to an oxidation state of 0 of nitrogen as reference
γi can be defined in another way by:
- Taking another oxidation state for each atom as reference
- Using other compounds in which the atoms are available to compare the oxidation state of the
atoms with the reference
Other definition of γi? -> other formula for calculating γi
Linear combination of atom balances (in which some rates are eliminated) can be made using the
echelon method:
Result of linear combination of atom balances = electron balance in which values of γ are known ->
electron balance is NOT an independent equation because it is a linear combination of the atom
balances, so it cannot be used as extra equation!
,Luedeking-Piret’s law: law for calculating the rate of product formation by a cell.
YPX = theoretical yield of product on cells
mP = maintenance constant -> measure for how much product is produced as a result of
maintenance.
Derivation of Luedeking-Piret’s law:
Example of electron balance when certain fermentation product is produced by cell:
rP = production rate of fermentation product
Rearranging this electron balance:
Substitute Pirt's law for rS:
Rearranging yields Luedeking-Piret’s law:
The formula above is true when the nitrogen source is NH3 and thus γN = 0, when another nitrogen
source is used and thus γN ≠ 0 the term γNrN doesn’t cancel in the electron balance and thus the term
γN
rN is included in the rearranged electron balance, so in the formula above also an extra term is
γP
included.
Cells -> never only produce fermentation product out of sugar:
- Always also produce CO2 out of sugar
- Use H2 or formate as extra electron sink (so besides fermentation product) if
necessary -> this type of metabolic information is often necessary to unravel So this is NEVER true!
reaction stoichiometry
Products produced by cells can be divided into 2 classes:
1. Necessary products: products which will always be produced by cells because their production is
necessary for the survival of the cells.
- Cell components (biomass)
- Catabolites (e.g. CO2, H2O, ethanol)
2. Luxury products: products which will NOT always be produced by cells because they are not
needed for primary metabolism and thus not necessary for the survival of the cells (e.g. penicillin,
citric acid).
2 ways of writing Pirt’s law:
1. Pirt’s law for necessary products:
,2. Pirt’s law when there are also luxury products -> extra term which is the energy which is needed
for the production of luxury products is included (in red)
qP (μ) models (models in which qP is described as a function of μ):
1. qP (μ) model which is found by rearranging Luedeking-Piret’s law a bit:
- For necessary products:
- When there are also luxury products:
2. qP (μ) model for parabolic repression by luxury product:
Combination of Pirt’s law for luxury products + Luedeking-Piret’s law:
mS = real maintenance which is necessary to keep the cell alive
𝑚𝑃
= no real maintenance, but constant specific uptake rate of energy source (sugar) for making
𝑌𝑃𝑆
product
Lecture 2 – Kinetics + chemostats
Reaction kinetics
For reaction kinetics there is no universal model, many different models can be found in literature
because of the different data obtained in different experiments on different cells:
,Specific rates (q’s) are in reality determined by many different factors (e.g. q = f(cS, cP, cW, T, …)), to
be able to use less sophisticated math and thus to simplify the assumption is made that one
component (reactant or product) is the rate-limiting component: the component (reactant or
product) which is least available and therefore determines the reaction rate.
2 possible approaches in using reaction kinetics to determine unknowns:
1. Approach 1:
- Use kinetic law to calculate specific growth rate (μ)
- Use this calculated μ to calculate specific rate of sugar uptake (qS) using Pirt’s law:
2. Approach 2:
- Use kinetic law to calculate specific rate of sugar uptake (q S)
- Use this calculated qS to calculate specific growth rate (μ) using Pirt’s law:
2 situations which often occur:
1. Sugar = rate-limiting component -> qS = f(cS)
2. Product = rate-limiting component (product inhibition) -> qP = g(cP)
1. Sugar (or another reactant) is rate-limiting component
Kinetic law to use = Monod’s law:
qS = specific rate of sugar consumption
qSMAX = maximum specific rate of sugar consumption -> y-asymptote of Monod-
curve
cS = concentration of sugar outside the cells -> this can also be the concentration of another reactant
instead of sugar, when you want to apply Monod’s law to another reactant than sugar
KS = Monod constant: sugar concentration (cS) at which a cell works at half of its maximum sugar
converting ability:
2. Product is rate-limiting component (product inhibition)
Kinetic law to use = Han-Levenspiel model -> can be written
for μ, qP, qS, mS etc. -> for example:
μMAX = maximum specific growth rate of cells
cP = product concentration outside the cells
,cPMAX = maximum product concentration outside the cells
qP = specific rate of product production
qPMAX = maximum specific rate of product production
Chemostats
Chemostat = “chemical composition static”
Chemostat: bioreactor which operates in steady state (state in which no parameters change during
the time) because:
- Medium is pumped continuously through the bioreactor
- There are no cells in the ingoing medium
- It is ideally mixed -> the concentration of a specific component is identical at every location inside
the bioreactor & concentration of component in outlet = concentration of component in bioreactor
Advantage of chemostat = cells are in an environment which is exactly defined by you
Disadvantage of chemostat = it takes time for the chemostat to reach steady state, so you have to
wait until you can start your chemostat experiment
Component balance for component i over the water in a bioreactor:
Chemostat?
- ρL,OUT = ρL (ideally mixed)
- Accumulation rate = 0 (continously operated)
- Transfer rate = 0 -> transfer rate only has to be included in component balances of gases or in
overall mass balance when there are gases present in the reactor
Overall mass balance chemostat (mass balance for all components in the chemostat together):
φL,INρL,IN-φLρL = 0
On laboratory scale (but definetily NOT on industrial scale) the following is even true for a
chemostat:
,Component balances over chemostat:
1. Component balance product/product balance:
Every other component balance for a chemostat can be derived using the same approach as used for
this component balance!
Chemostats -> dilution rate (DL):
φ
DL =
VL
φ = volumetric flow rate in chemostat
VL = volume of the liquid inside the chemostat
2. Component balance cells/cell balance:
cXIN = 0 for chemostat -> –DLcX + μcX = 0
So in a chemostat you can choose μ by choosing DL!
Model for chemostat must include:
1. All following component balances:
- Component balance of desired product
- Component balance of cells
- Component balance of rate-limiting component
2. Kinetic law:
- Reactant (e.g. sugar) is rate-limiting component -> Monod’s law
- Product is rate-limiting component (product inhibition) -> Han-Levenspiel model
3. Pirt’s law
4. Luedeking-Piret’s law -> for determining stoichiometry luxury products
5. Atom balances and/or electron balance -> for determining stoichiometry necessary products (e.g.
catabolites: CO2, H2O, fermentation products such as ethanol)
Finding parameters of Monod’s law and Pirt’s law:
1. Choose DL and cSIN
2. Measure cS and cX
3. Calculate μ from cell balance and qS from sugar balance:
4. 2 possibilities for determining parameters of Monod’s law and Pirt’s law:
- Determine the parameters by algebraically solving:
> Monod’s law -> fill in the different values of cS and qS which are obtained at 2 different DL’s and
solve the resulting system of 2 equations algebraically for qSMAX and KS (the parameters of Monod’s
law)
,> Pirt’s law -> fill in the different values of qS and μ which are obtained at 2 different DL’s and solve
the resulting system of 2 equations algebraically for YXS and mS (the parameters of Pirt’s law)
- Determine the parameters by linear regression:
> Monod’s law -> Lineweaver-Burke method: linearizing Monod’s law by raising the left and the right
hand side of Monod’s law both to the power -1 and subsequently plotting 1/qS on the y-axis against
1/cS on the x-axis to determine the parameters of Monod’s law (qSMAX and KS):
= intercept with y-axis
= slope of graph
This method can be inaccurate if there are not enough measurements done at high cS and there are
big errors at low cS, so when applying this method attention should be paid that enough
measurement (high and low) of cS are made
> Pirt’s law -> no linearisation necessary because Pirt’s law is already linear
-> Pirt’s law for necessary products (there are no
luxury products)
= intercept with y-axis
= slope of graph
, Design with known parameters of Monod’s law and Pirt’s law:
So in chemostat cS is also independent of cSIN, because cSIN is not in one of the equations above.
Washout: the dilution rate (D) is too high, so cells are washed out of the chemostat. -> cX approaches
0, so cS approaches cSIN (less cells in chemostat, so less sugar is converted)
Washout occurs when D ≥ μ, so:
- Sugar = rate-limiting? -> Monod’s law:
D ≥ μMAX * cS/(KS + cS) with cS = cSIN
- Product = rate-limiting? -> Han-Levenspiel model:
D μMAX * (1 – cP/cPMAX) with cP = cPIN
D < μ -> chemostat reaches steady state with cells (no washout)
D ≥ μ -> chemostat reaches steady state without cells (washout has occurred)
Han-Levenspiel model -> two different forms:
1. For μ:
Lecture 1 - Stoichiometry
2 types of models for cells:
1. Black box model: model in which a cell is approached as a black box, so the metabolic reactions
occurring inside the cell are unknown and only the result of this reactions is known.
2. Metabolic pathway model: model in which all metabolic reactions occurring inside the cell are
known.
Past -> cell was discovered, intensively studied and then it was tried to apply the cell in a process
Nowadays -> work from process to cell, so there is a process and we look for (possibly adapted) cells
to fit in this process
Reaction which occurs inside cell is described by stoichiometric reaction equation:
rS CH2O + rN nitrogen source + rO electron acceptor + rX cells/biomass + rC CO2 + rW H2O = 0
Remarks for the stoichiometric reaction equation:
1. Reaction equation is written as a balance, so without
2. r:
r > 0 -> r = production rate CH2O = sugar (in Cmoles)
r < 0 -> r = consumption rate
So for reactants r is negative and for products r is positive
3. For simplicity no ions but only neutral compounds are included
4. Cmoles are used for all C-containing compounds, normal moles are used for compounds without
C-atoms -> so molecular formula of all C-containing compounds is written in a way that it contains 1
C-atom
5. Usually:
Nitrogen source = NH3 (ammonia)
Electron acceptor = O2 (oxygen)
Cells/biomass = CH2O0.5N0.2 (typical cell composition)
2 types of rates:
1. Volumetric rate (ri) -> [ri] = Cmolei m-3 s-1
2. Specific rate (qi) -> [qi] = Cmolei Cmolex-1 s-1
- For products -> + qi
- For reactants -> - qi
- For cells -> qx = μ (specific growth rate) -> rX = μcX
cX = concentration of cells/biomass
Finding reaction stoichiometry/all reaction rates:
1. Atom balances
2. Electron balance = shortcut to find consumption rate of electron acceptor. -> not necessarily used,
because consumption rate of electron acceptor can also be found using the atom balances
3. Pirt’s law (if you don’t have enough equations and thus need an extra equation)
,Atom balances: balances for all types of atoms which occur in the reaction equation.
Sugar (CH2O) -> used in cells for two purposes at the same time (parallel reactions):
1. Cell growth/making new cells -> by using sugar as carbon source
2. Maintain cells -> by using sugar as energy source (ATP produced using the sugar, ATP is used for
maintenance of cells)
Pirt’s law: law which describes consumption rate of sugar (rS) by cells by telling for what sugar is used
by the cells.
YXS = theoretical yield
mS = maintenance coefficient } -> determined with measurements or estimated using metabolic model
cX = concentration of cells -> can be chosen when designing process:
Higher cX? -> easier and thus cheaper DSP (downstream processing), but more sugar + electron
acceptor needed
Lower cX? -> less easy and thus less cheap DSP (downstream processing), but less sugar + electron
acceptor needed
2 types of yield which should be distinguished from each other:
1. Theoretical yield (YXS): yield which can be calculated from Pirt’s law (for Y XS) or Luedeking-Piret’s
law (for YPX), but which will never be measured in reality.
2. Observed yield (YXSOBS): yield which is actually measured in reality.
Observed yield -> defined as ratio of 2 production rates:
YXSOBS = rX/-rS (minus sign is necessary to make sure that YXSOBS is a positive number) = qX/-qS = μ/-qS
YPXOBS = rP/rX = qP/qX = qP/μ
Theoretical yield (YXS) is always bigger than the observed yield (YXSOBS), because in reality sugar is
besides for production of biomass also used for maintenance of the cells
Theoretical yield (YPX) is always smaller than the observed yield (YPXOBS), because in reality there is
also maintenance of the cells and not only biomass production -> products are also produced during
maintenance and not only during biomass production
Electron balance: balance in which the electrons going into the reaction are compared to the
electrons coming out of the reaction. -> using electron balance is useful as shortcut to find
consumption rate of electron acceptor
,Reaction equation:
Corresponding electron balance:
Electron balance can also be made
using q (specific rate) instead of r -> ri
γN, γC and γW are zero, so the electron balance becomes: in electron balance to the left is
replaced by corresponding qi
γi = degree of reduction of component i: number of electrons which is
available in component i when compared to its most oxidized form (e.g. CO2 for carbon components).
Explanation of terms:
4 * #C -> oxidation state of C in CO2 (compound in which C is most oxidized) is +4, so 4 electrons are
available by every carbon atom when γ is defined relative to an oxidation state of 0 of carbon as
reference
1 * #H -> oxidation state of H in H+ (compound in which H is most oxidized) is +1, so 1 electron is
available by every hydrogen atom when γ is defined relative to an oxidation state of 0 of hydrogen as
reference
-2 * #O -> oxidation state of O in H2O is -2, so -2 electrons are available by every oxygen atom when γ
is defined relative to an oxidation state of 0 of oxygen as reference
-3 * #N -> oxidation state of N in NH3 is -3, so -3 electrons are available by every nitrogen atom when
γ is defined relative to an oxidation state of 0 of nitrogen as reference
γi can be defined in another way by:
- Taking another oxidation state for each atom as reference
- Using other compounds in which the atoms are available to compare the oxidation state of the
atoms with the reference
Other definition of γi? -> other formula for calculating γi
Linear combination of atom balances (in which some rates are eliminated) can be made using the
echelon method:
Result of linear combination of atom balances = electron balance in which values of γ are known ->
electron balance is NOT an independent equation because it is a linear combination of the atom
balances, so it cannot be used as extra equation!
,Luedeking-Piret’s law: law for calculating the rate of product formation by a cell.
YPX = theoretical yield of product on cells
mP = maintenance constant -> measure for how much product is produced as a result of
maintenance.
Derivation of Luedeking-Piret’s law:
Example of electron balance when certain fermentation product is produced by cell:
rP = production rate of fermentation product
Rearranging this electron balance:
Substitute Pirt's law for rS:
Rearranging yields Luedeking-Piret’s law:
The formula above is true when the nitrogen source is NH3 and thus γN = 0, when another nitrogen
source is used and thus γN ≠ 0 the term γNrN doesn’t cancel in the electron balance and thus the term
γN
rN is included in the rearranged electron balance, so in the formula above also an extra term is
γP
included.
Cells -> never only produce fermentation product out of sugar:
- Always also produce CO2 out of sugar
- Use H2 or formate as extra electron sink (so besides fermentation product) if
necessary -> this type of metabolic information is often necessary to unravel So this is NEVER true!
reaction stoichiometry
Products produced by cells can be divided into 2 classes:
1. Necessary products: products which will always be produced by cells because their production is
necessary for the survival of the cells.
- Cell components (biomass)
- Catabolites (e.g. CO2, H2O, ethanol)
2. Luxury products: products which will NOT always be produced by cells because they are not
needed for primary metabolism and thus not necessary for the survival of the cells (e.g. penicillin,
citric acid).
2 ways of writing Pirt’s law:
1. Pirt’s law for necessary products:
,2. Pirt’s law when there are also luxury products -> extra term which is the energy which is needed
for the production of luxury products is included (in red)
qP (μ) models (models in which qP is described as a function of μ):
1. qP (μ) model which is found by rearranging Luedeking-Piret’s law a bit:
- For necessary products:
- When there are also luxury products:
2. qP (μ) model for parabolic repression by luxury product:
Combination of Pirt’s law for luxury products + Luedeking-Piret’s law:
mS = real maintenance which is necessary to keep the cell alive
𝑚𝑃
= no real maintenance, but constant specific uptake rate of energy source (sugar) for making
𝑌𝑃𝑆
product
Lecture 2 – Kinetics + chemostats
Reaction kinetics
For reaction kinetics there is no universal model, many different models can be found in literature
because of the different data obtained in different experiments on different cells:
,Specific rates (q’s) are in reality determined by many different factors (e.g. q = f(cS, cP, cW, T, …)), to
be able to use less sophisticated math and thus to simplify the assumption is made that one
component (reactant or product) is the rate-limiting component: the component (reactant or
product) which is least available and therefore determines the reaction rate.
2 possible approaches in using reaction kinetics to determine unknowns:
1. Approach 1:
- Use kinetic law to calculate specific growth rate (μ)
- Use this calculated μ to calculate specific rate of sugar uptake (qS) using Pirt’s law:
2. Approach 2:
- Use kinetic law to calculate specific rate of sugar uptake (q S)
- Use this calculated qS to calculate specific growth rate (μ) using Pirt’s law:
2 situations which often occur:
1. Sugar = rate-limiting component -> qS = f(cS)
2. Product = rate-limiting component (product inhibition) -> qP = g(cP)
1. Sugar (or another reactant) is rate-limiting component
Kinetic law to use = Monod’s law:
qS = specific rate of sugar consumption
qSMAX = maximum specific rate of sugar consumption -> y-asymptote of Monod-
curve
cS = concentration of sugar outside the cells -> this can also be the concentration of another reactant
instead of sugar, when you want to apply Monod’s law to another reactant than sugar
KS = Monod constant: sugar concentration (cS) at which a cell works at half of its maximum sugar
converting ability:
2. Product is rate-limiting component (product inhibition)
Kinetic law to use = Han-Levenspiel model -> can be written
for μ, qP, qS, mS etc. -> for example:
μMAX = maximum specific growth rate of cells
cP = product concentration outside the cells
,cPMAX = maximum product concentration outside the cells
qP = specific rate of product production
qPMAX = maximum specific rate of product production
Chemostats
Chemostat = “chemical composition static”
Chemostat: bioreactor which operates in steady state (state in which no parameters change during
the time) because:
- Medium is pumped continuously through the bioreactor
- There are no cells in the ingoing medium
- It is ideally mixed -> the concentration of a specific component is identical at every location inside
the bioreactor & concentration of component in outlet = concentration of component in bioreactor
Advantage of chemostat = cells are in an environment which is exactly defined by you
Disadvantage of chemostat = it takes time for the chemostat to reach steady state, so you have to
wait until you can start your chemostat experiment
Component balance for component i over the water in a bioreactor:
Chemostat?
- ρL,OUT = ρL (ideally mixed)
- Accumulation rate = 0 (continously operated)
- Transfer rate = 0 -> transfer rate only has to be included in component balances of gases or in
overall mass balance when there are gases present in the reactor
Overall mass balance chemostat (mass balance for all components in the chemostat together):
φL,INρL,IN-φLρL = 0
On laboratory scale (but definetily NOT on industrial scale) the following is even true for a
chemostat:
,Component balances over chemostat:
1. Component balance product/product balance:
Every other component balance for a chemostat can be derived using the same approach as used for
this component balance!
Chemostats -> dilution rate (DL):
φ
DL =
VL
φ = volumetric flow rate in chemostat
VL = volume of the liquid inside the chemostat
2. Component balance cells/cell balance:
cXIN = 0 for chemostat -> –DLcX + μcX = 0
So in a chemostat you can choose μ by choosing DL!
Model for chemostat must include:
1. All following component balances:
- Component balance of desired product
- Component balance of cells
- Component balance of rate-limiting component
2. Kinetic law:
- Reactant (e.g. sugar) is rate-limiting component -> Monod’s law
- Product is rate-limiting component (product inhibition) -> Han-Levenspiel model
3. Pirt’s law
4. Luedeking-Piret’s law -> for determining stoichiometry luxury products
5. Atom balances and/or electron balance -> for determining stoichiometry necessary products (e.g.
catabolites: CO2, H2O, fermentation products such as ethanol)
Finding parameters of Monod’s law and Pirt’s law:
1. Choose DL and cSIN
2. Measure cS and cX
3. Calculate μ from cell balance and qS from sugar balance:
4. 2 possibilities for determining parameters of Monod’s law and Pirt’s law:
- Determine the parameters by algebraically solving:
> Monod’s law -> fill in the different values of cS and qS which are obtained at 2 different DL’s and
solve the resulting system of 2 equations algebraically for qSMAX and KS (the parameters of Monod’s
law)
,> Pirt’s law -> fill in the different values of qS and μ which are obtained at 2 different DL’s and solve
the resulting system of 2 equations algebraically for YXS and mS (the parameters of Pirt’s law)
- Determine the parameters by linear regression:
> Monod’s law -> Lineweaver-Burke method: linearizing Monod’s law by raising the left and the right
hand side of Monod’s law both to the power -1 and subsequently plotting 1/qS on the y-axis against
1/cS on the x-axis to determine the parameters of Monod’s law (qSMAX and KS):
= intercept with y-axis
= slope of graph
This method can be inaccurate if there are not enough measurements done at high cS and there are
big errors at low cS, so when applying this method attention should be paid that enough
measurement (high and low) of cS are made
> Pirt’s law -> no linearisation necessary because Pirt’s law is already linear
-> Pirt’s law for necessary products (there are no
luxury products)
= intercept with y-axis
= slope of graph
, Design with known parameters of Monod’s law and Pirt’s law:
So in chemostat cS is also independent of cSIN, because cSIN is not in one of the equations above.
Washout: the dilution rate (D) is too high, so cells are washed out of the chemostat. -> cX approaches
0, so cS approaches cSIN (less cells in chemostat, so less sugar is converted)
Washout occurs when D ≥ μ, so:
- Sugar = rate-limiting? -> Monod’s law:
D ≥ μMAX * cS/(KS + cS) with cS = cSIN
- Product = rate-limiting? -> Han-Levenspiel model:
D μMAX * (1 – cP/cPMAX) with cP = cPIN
D < μ -> chemostat reaches steady state with cells (no washout)
D ≥ μ -> chemostat reaches steady state without cells (washout has occurred)
Han-Levenspiel model -> two different forms:
1. For μ:
