Enzyme Catalysis and Microbial Growth Kinetics

Purpose. Construct and interrogate growth, inhibition, maintenance, and product-rate models; discriminate identifiable parameters from fitted numbers.

Entry knowledge. Students should be able to balance mass and energy, work with differential equations, check dimensional consistency and interpret experimental uncertainty. Prior biological experience helps but is not assumed. The models are deliberately simplified; scientific claims require the limitations in each lesson to be stated.

Lesson pathway

1. [Monod growth and nutrient limitation](../lessons/unit-02-lesson-01.html) — [related numerical lab](../labs/lab-04.html)
2. [Substrate inhibition, maintenance and death](../lessons/unit-02-lesson-02.html) — [related numerical lab](../labs/lab-05.html)
3. [Product kinetics and non-growth-associated synthesis](../lessons/unit-02-lesson-03.html) — [related numerical lab](../labs/lab-06.html)
4. [Kinetic fitting, identifiability and extrapolation](../lessons/unit-02-lesson-04.html) — [related numerical lab](../labs/lab-06.html)

Practise and demonstrate competence

• [Unit 02 assessment: 16 multi-part tasks](../assessments/unit-02.html) tests physical reasoning, calculation, falsification and design decision-making.
• [Unit 02 applied project](../projects/unit-02.html) requires an evidence ledger, assumption register, data table, quantitative check and defensible conclusion.
• [Laboratory manual](../LABORATORY-MANUAL.html) indexes the offline simulations and their modeling restrictions.

Outcomes and publication boundary

At the end of this unit, students should explain why the chosen equation is appropriate, reproduce at least one numerical prediction, describe a test that could disprove it, and articulate the information missing for a real process decision. No numerical curve in this resource is an industrial operating recommendation. All sample coefficients and operating points are fictional classroom inputs unless a source is specifically identified.

Unit 2 · Lesson 1 — Monod growth and nutrient limitation

[Unit 02 overview](../units/unit-02.html) · [Run numerical laboratory 04](../labs/lab-04.html) · [Assessment](../assessments/unit-02.html) · [Applied project](../projects/unit-02.html)

The engineering question

The Monod equation represents the dependence of specific growth rate on a chosen limiting substrate. It is analogous in shape to Michaelis–Menten kinetics but is a population model: Ks is not automatically an enzyme dissociation constant. For a batch culture, a substrate balance must accompany the growth equation; otherwise an unconstrained growth curve can create mass without consuming food. Substrate concentration can be low while growth remains vigorous if transport is efficient, but a different nutrient or oxygen may become limiting before the modeled nutrient is depleted. Parameter estimates depend on the range of measured substrate concentrations.

The larger aim of Biokinetics is to answer this question: Construct and interrogate growth, inhibition, maintenance, and product-rate models; discriminate identifiable parameters from fitted numbers. A result must be interpreted as a statement about a specified physical boundary and observed quantities. Distinguish the underlying biological mechanism from the mathematical convenience that makes a calculation tractable. A model can be dimensionally consistent and still be badly chosen for the organism, medium or scale. Conversely, a complicated simulator is not superior unless its extra parameters can be identified and tested with observations.

A useful model and its assumptions

Working relation: μ(S)=μmax S/(Ks+S), dX/dt=μX and dS/dt=−μX/Yxs when maintenance is omitted. Biomass yield Yxs has units g biomass/g substrate.

The displayed relationship is a starting point for reasoning. Before substituting numbers, name every term, give the unit and decide whether each quantity is a measured value, a fitted coefficient, or a controlled setting. Conservation laws are independent of constitutive assumptions, while kinetic and transfer coefficients depend on conditions. Make a sketch of the relevant system boundary and show what enters, accumulates and leaves. State whether the rate is per unit liquid volume, per unit catalyst or per unit cell mass; confusing these bases can lead to order-of-magnitude errors even when individual values seem realistic.

When a relationship contains an exponential, check which variable is integrated in the exponent and whether the exponent is dimensionless. When it contains a ratio, check what happens as the denominator approaches a limiting condition. When it describes an observable signal, identify the calibration that connects that signal to the claimed physical quantity. These checks are not substitutes for experiment, but they reject many wrong models before fitting begins.

Worked investigation with synthetic numbers

For μmax=0.5 h⁻¹, Ks=0.8 g/L and S=1.2 g/L, μ=0.3 h⁻¹. A batch with X=0.4 g/L and yield 0.5 g/g initially consumes 0.24 g substrate/L/h. That instantaneous rate cannot be maintained without checking the declining S.

Reconstruction task. Rewrite the stated computation as a ledger: input variable and unit, conversion used, intermediate result and final quantity. Show the same calculation with one input perturbed by 10% while retaining the others, and determine whether the output response is approximately proportional, nonlinear, or sensitive to a constraint. Then reverse the question: what input accuracy would be needed to distinguish the two predictions using a real instrument? Avoid claiming more significant figures than the chosen coefficients justify.

The example is intentionally synthetic. It should teach the direction and scale of a response, not imply that a specific organism or process will exhibit these numerical coefficients. If the result suggests substantial accumulation, ask where the required mass and energy originate. If it suggests extreme sensitivity, inspect whether the sensitivity arises from a genuine mechanistic feature or from mathematically approaching a singular assumption.

Measurement design and engineering judgment

Design a challenge experiment that can fail the model. Use concentrations spanning low and high substrate response, include controls, and test whether the fitted function predicts a held-out condition. Record pH, temperature and the assay method as part of the model definition.

Estimate μmax and Ks jointly with confidence intervals across low and high substrate conditions. Inspect whether growth-rate observations at small S exceed the measurement error.

A defensible evidence plan identifies a process objective, measured signals, their frequency, instrument or assay uncertainty, and a comparison criterion selected before viewing the data. Include control conditions and independent replicates when the biological material varies across runs. Distinguish calibration measurements from hold-out verification data. If the main parameter is estimated from one narrow operating window, do not assume it will remain identifiable when oxygen, temperature, shear, pH or medium composition changes.

In a design review, explain the alternatives rather than selecting a model because it yields the highest projected titer or lowest cost. A common problem is conflating process capacity with process utilization: an available transfer rate is not automatically used, and biological demand can be low because the culture is inhibited. When a prediction is contradicted, investigate both the experiment and the model, and keep a record of the resulting revision.

Where the simple model fails

The model omits substrate inhibition, maintenance demand, cell death and transport heterogeneity. A fitted Ks may absorb oxygen limitation rather than represent nutrient affinity.

A classroom model is especially fragile when extrapolated beyond its parameter-identification range. Carefully distinguish structural uncertainty (missing mechanism or wrong equation), parameter uncertainty (imprecisely estimated coefficient), and measurement uncertainty (sensor or assay error). Running many simulations with one incorrect equation does not resolve structural uncertainty. A useful sensitivity chart should therefore accompany a stated model boundary, not pretend to replace it.

For an independent verification, perform three checks: units and conservation; a limiting condition with a known qualitative answer; and a second calculation method or external measurement. Explain what each check can and cannot demonstrate. Automated agreement with an equation confirms the implementation, not necessarily the equation's suitability for a real reactor.

Guided laboratory and discussion

Open [Laboratory 04](../labs/lab-04.html). Record the default input values with units. Move the first slider to three equally spaced settings while leaving other controls unchanged; save the response to CSV and compare the graph with the governing relation in that laboratory. Then change a second independent parameter to test whether the original trend is stable. The laboratory describes its physical limits and reports an explicit warning where assumptions fail. The export is generated locally; no data leave the browser.

Investigative prompt: Develop an explicit-Euler time-step experiment and compare with a smaller time step to detect numerical artifacts.

Interpretation prompt: Identify a response that the model cannot predict, and propose one targeted measurement needed to decide if the omission matters. Write a two-sentence decision: what is supported by the simulation and what remains unproven. A high-grade response distinguishes a mathematical identity, an observation and a contingent engineering inference.

Lesson check

1. State the modeled system boundary, the type of measurable quantity being predicted and the appropriate units.
2. Derive or reconstruct the stated relation from a rate law or conservation balance, and show a limiting case.
3. Reproduce the worked calculation, then vary one input and interpret sensitivity without generalizing outside the stated model.
4. Identify two independent measurements that could falsify the simplified hypothesis.
5. Present a recommendation that is explicitly conditional on the evidence rather than an unsupported production specification.

Continue: Substrate inhibition, maintenance and death. Return to the [unit learning pathway](../units/unit-02.html) or attempt the [unit assessment](../assessments/unit-02.html).

Unit 2 · Lesson 2 — Substrate inhibition, maintenance and death

[Unit 02 overview](../units/unit-02.html) · [Run numerical laboratory 05](../labs/lab-05.html) · [Assessment](../assessments/unit-02.html) · [Applied project](../projects/unit-02.html)

The engineering question

Increasing substrate can depress net growth when toxicity, osmotic stress or regulatory inhibition dominates. The Haldane/Andrews empirical form introduces a high-substrate inhibition term and therefore predicts a growth-rate maximum at a finite concentration. Maintenance consumption is different: cells can burn substrate without increasing biomass. A kinetic model that assumes every gram of substrate consumed becomes biomass or growth-associated product will overstate yield when maintenance is substantial. Cell death introduces yet another observable, viable biomass, that may differ from total dry mass.

The larger aim of Biokinetics is to answer this question: Construct and interrogate growth, inhibition, maintenance, and product-rate models; discriminate identifiable parameters from fitted numbers. A result must be interpreted as a statement about a specified physical boundary and observed quantities. Distinguish the underlying biological mechanism from the mathematical convenience that makes a calculation tractable. A model can be dimensionally consistent and still be badly chosen for the organism, medium or scale. Conversely, a complicated simulator is not superior unless its extra parameters can be identified and tested with observations.

A useful model and its assumptions

Working relation: A common empirical form is μ=μmax S/(Ks+S+S²/Ki). A maintenance-inclusive uptake model is qS=μ/Yxs+mS; units of qS and mS are g substrate per g biomass per hour.

The displayed relationship is a starting point for reasoning. Before substituting numbers, name every term, give the unit and decide whether each quantity is a measured value, a fitted coefficient, or a controlled setting. Conservation laws are independent of constitutive assumptions, while kinetic and transfer coefficients depend on conditions. Make a sketch of the relevant system boundary and show what enters, accumulates and leaves. State whether the rate is per unit liquid volume, per unit catalyst or per unit cell mass; confusing these bases can lead to order-of-magnitude errors even when individual values seem realistic.

When a relationship contains an exponential, check which variable is integrated in the exponent and whether the exponent is dimensionless. When it contains a ratio, check what happens as the denominator approaches a limiting condition. When it describes an observable signal, identify the calibration that connects that signal to the claimed physical quantity. These checks are not substitutes for experiment, but they reject many wrong models before fitting begins.

Worked investigation with synthetic numbers

Let μmax=0.6 h⁻¹, Ks=0.4 g/L, Ki=20 g/L. At S=2 g/L, μ≈0.462 h⁻¹; at S=20 g/L, μ≈0.297 h⁻¹. More food reduces predicted growth under the model. Adding mS=0.03 g/g/h raises the uptake prediction even if μ stays unchanged.

Reconstruction task. Rewrite the stated computation as a ledger: input variable and unit, conversion used, intermediate result and final quantity. Show the same calculation with one input perturbed by 10% while retaining the others, and determine whether the output response is approximately proportional, nonlinear, or sensitive to a constraint. Then reverse the question: what input accuracy would be needed to distinguish the two predictions using a real instrument? Avoid claiming more significant figures than the chosen coefficients justify.

The example is intentionally synthetic. It should teach the direction and scale of a response, not imply that a specific organism or process will exhibit these numerical coefficients. If the result suggests substantial accumulation, ask where the required mass and energy originate. If it suggests extreme sensitivity, inspect whether the sensitivity arises from a genuine mechanistic feature or from mathematically approaching a singular assumption.

Measurement design and engineering judgment

Design a challenge experiment that can fail the model. Use concentrations spanning low and high substrate response, include controls, and test whether the fitted function predicts a held-out condition. Record pH, temperature and the assay method as part of the model definition.

Collect high-concentration observations rather than fitting inhibition using low-S data only. Compare alternative hypotheses: osmolarity, dissolved oxygen and assay bias.

A defensible evidence plan identifies a process objective, measured signals, their frequency, instrument or assay uncertainty, and a comparison criterion selected before viewing the data. Include control conditions and independent replicates when the biological material varies across runs. Distinguish calibration measurements from hold-out verification data. If the main parameter is estimated from one narrow operating window, do not assume it will remain identifiable when oxygen, temperature, shear, pH or medium composition changes.

In a design review, explain the alternatives rather than selecting a model because it yields the highest projected titer or lowest cost. A common problem is conflating process capacity with process utilization: an available transfer rate is not automatically used, and biological demand can be low because the culture is inhibited. When a prediction is contradicted, investigate both the experiment and the model, and keep a record of the resulting revision.

Where the simple model fails

A quadratic inhibition denominator is empirical and may fail outside its calibration range. The same decline in observed viable cells can arise from death or reduced specific growth.

A classroom model is especially fragile when extrapolated beyond its parameter-identification range. Carefully distinguish structural uncertainty (missing mechanism or wrong equation), parameter uncertainty (imprecisely estimated coefficient), and measurement uncertainty (sensor or assay error). Running many simulations with one incorrect equation does not resolve structural uncertainty. A useful sensitivity chart should therefore accompany a stated model boundary, not pretend to replace it.

For an independent verification, perform three checks: units and conservation; a limiting condition with a known qualitative answer; and a second calculation method or external measurement. Explain what each check can and cannot demonstrate. Automated agreement with an equation confirms the implementation, not necessarily the equation's suitability for a real reactor.

Guided laboratory and discussion

Open [Laboratory 05](../labs/lab-05.html). Record the default input values with units. Move the first slider to three equally spaced settings while leaving other controls unchanged; save the response to CSV and compare the graph with the governing relation in that laboratory. Then change a second independent parameter to test whether the original trend is stable. The laboratory describes its physical limits and reports an explicit warning where assumptions fail. The export is generated locally; no data leave the browser.

Investigative prompt: Choose a feeding window that avoids inhibitory concentrations while maintaining the desired productivity, and quantify sensitivity to Ki.

Interpretation prompt: Identify a response that the model cannot predict, and propose one targeted measurement needed to decide if the omission matters. Write a two-sentence decision: what is supported by the simulation and what remains unproven. A high-grade response distinguishes a mathematical identity, an observation and a contingent engineering inference.

Lesson check

1. State the modeled system boundary, the type of measurable quantity being predicted and the appropriate units.
2. Derive or reconstruct the stated relation from a rate law or conservation balance, and show a limiting case.
3. Reproduce the worked calculation, then vary one input and interpret sensitivity without generalizing outside the stated model.
4. Identify two independent measurements that could falsify the simplified hypothesis.
5. Present a recommendation that is explicitly conditional on the evidence rather than an unsupported production specification.

Continue: Product kinetics and non-growth-associated synthesis. Return to the [unit learning pathway](../units/unit-02.html) or attempt the [unit assessment](../assessments/unit-02.html).

Unit 2 · Lesson 3 — Product kinetics and non-growth-associated synthesis

[Unit 02 overview](../units/unit-02.html) · [Run numerical laboratory 06](../labs/lab-06.html) · [Assessment](../assessments/unit-02.html) · [Applied project](../projects/unit-02.html)

The engineering question

A product can track biomass formation, biomass presence, or both. The Luedeking–Piret relation separates a term proportional to growth from one proportional to viable biomass. For an extracellular metabolite, an additional transport or degradation term can be required. A production rate normalized per reactor volume is not the same as a cell-specific production rate; dilution from feeding changes concentration even if total product mass rises. Product may accumulate during stationary phase with little new biomass, but that does not automatically establish a constant β across the culture life.

The larger aim of Biokinetics is to answer this question: Construct and interrogate growth, inhibition, maintenance, and product-rate models; discriminate identifiable parameters from fitted numbers. A result must be interpreted as a statement about a specified physical boundary and observed quantities. Distinguish the underlying biological mechanism from the mathematical convenience that makes a calculation tractable. A model can be dimensionally consistent and still be badly chosen for the organism, medium or scale. Conversely, a complicated simulator is not superior unless its extra parameters can be identified and tested with observations.

A useful model and its assumptions

Working relation: dP/dt = α dX/dt + β X is a simplified growth-associated plus non-growth-associated production relation for a constant-volume culture without product loss. α is g product/g biomass; β is g product/(g biomass·h).

The displayed relationship is a starting point for reasoning. Before substituting numbers, name every term, give the unit and decide whether each quantity is a measured value, a fitted coefficient, or a controlled setting. Conservation laws are independent of constitutive assumptions, while kinetic and transfer coefficients depend on conditions. Make a sketch of the relevant system boundary and show what enters, accumulates and leaves. State whether the rate is per unit liquid volume, per unit catalyst or per unit cell mass; confusing these bases can lead to order-of-magnitude errors even when individual values seem realistic.

When a relationship contains an exponential, check which variable is integrated in the exponent and whether the exponent is dimensionless. When it contains a ratio, check what happens as the denominator approaches a limiting condition. When it describes an observable signal, identify the calibration that connects that signal to the claimed physical quantity. These checks are not substitutes for experiment, but they reject many wrong models before fitting begins.

Worked investigation with synthetic numbers

With X₀=1 g/L, μ=0.12 h⁻¹ over 6 h, α=0.2 g/g and β=0.015 g/g/h, X≈2.05 g/L. Growth-associated product is ≈0.211 g/L, while the β contribution requires integrating X over time; using X₀ alone would underestimate it.

Reconstruction task. Rewrite the stated computation as a ledger: input variable and unit, conversion used, intermediate result and final quantity. Show the same calculation with one input perturbed by 10% while retaining the others, and determine whether the output response is approximately proportional, nonlinear, or sensitive to a constraint. Then reverse the question: what input accuracy would be needed to distinguish the two predictions using a real instrument? Avoid claiming more significant figures than the chosen coefficients justify.

The example is intentionally synthetic. It should teach the direction and scale of a response, not imply that a specific organism or process will exhibit these numerical coefficients. If the result suggests substantial accumulation, ask where the required mass and energy originate. If it suggests extreme sensitivity, inspect whether the sensitivity arises from a genuine mechanistic feature or from mathematically approaching a singular assumption.

Measurement design and engineering judgment

Design a challenge experiment that can fail the model. Use concentrations spanning low and high substrate response, include controls, and test whether the fitted function predicts a held-out condition. Record pH, temperature and the assay method as part of the model definition.

Fit product and biomass observations jointly, then test the result on a held-out portion of the batch. Report product assay detection range and whether the product degrades.

A defensible evidence plan identifies a process objective, measured signals, their frequency, instrument or assay uncertainty, and a comparison criterion selected before viewing the data. Include control conditions and independent replicates when the biological material varies across runs. Distinguish calibration measurements from hold-out verification data. If the main parameter is estimated from one narrow operating window, do not assume it will remain identifiable when oxygen, temperature, shear, pH or medium composition changes.

In a design review, explain the alternatives rather than selecting a model because it yields the highest projected titer or lowest cost. A common problem is conflating process capacity with process utilization: an available transfer rate is not automatically used, and biological demand can be low because the culture is inhibited. When a prediction is contradicted, investigate both the experiment and the model, and keep a record of the resulting revision.

Where the simple model fails

The law is an empirical approximation and does not reveal intracellular regulatory mechanisms. With fed-batch dilution, write mass balances on V·P rather than on P alone.

A classroom model is especially fragile when extrapolated beyond its parameter-identification range. Carefully distinguish structural uncertainty (missing mechanism or wrong equation), parameter uncertainty (imprecisely estimated coefficient), and measurement uncertainty (sensor or assay error). Running many simulations with one incorrect equation does not resolve structural uncertainty. A useful sensitivity chart should therefore accompany a stated model boundary, not pretend to replace it.

For an independent verification, perform three checks: units and conservation; a limiting condition with a known qualitative answer; and a second calculation method or external measurement. Explain what each check can and cannot demonstrate. Automated agreement with an equation confirms the implementation, not necessarily the equation's suitability for a real reactor.

Guided laboratory and discussion

Open [Laboratory 06](../labs/lab-06.html). Record the default input values with units. Move the first slider to three equally spaced settings while leaving other controls unchanged; save the response to CSV and compare the graph with the governing relation in that laboratory. Then change a second independent parameter to test whether the original trend is stable. The laboratory describes its physical limits and reports an explicit warning where assumptions fail. The export is generated locally; no data leave the browser.

Investigative prompt: Compare two hypothetical organisms with the same final titer but different productivity and substrate consumption and select a meaningful decision metric.

Interpretation prompt: Identify a response that the model cannot predict, and propose one targeted measurement needed to decide if the omission matters. Write a two-sentence decision: what is supported by the simulation and what remains unproven. A high-grade response distinguishes a mathematical identity, an observation and a contingent engineering inference.

Lesson check

1. State the modeled system boundary, the type of measurable quantity being predicted and the appropriate units.
2. Derive or reconstruct the stated relation from a rate law or conservation balance, and show a limiting case.
3. Reproduce the worked calculation, then vary one input and interpret sensitivity without generalizing outside the stated model.
4. Identify two independent measurements that could falsify the simplified hypothesis.
5. Present a recommendation that is explicitly conditional on the evidence rather than an unsupported production specification.

Continue: Kinetic fitting, identifiability and extrapolation. Return to the [unit learning pathway](../units/unit-02.html) or attempt the [unit assessment](../assessments/unit-02.html).

Unit 2 · Lesson 4 — Kinetic fitting, identifiability and extrapolation

[Unit 02 overview](../units/unit-02.html) · [Run numerical laboratory 06](../labs/lab-06.html) · [Assessment](../assessments/unit-02.html) · [Applied project](../projects/unit-02.html)

The engineering question

Parameter estimation asks whether data support a model and whether the individual parameters are distinguishable. Two sets of μmax and Ks can produce nearly identical Monod growth at concentrations far above Ks. Fitting rich high-S data alone often identifies μmax but leaves Ks uncertain. Confidence intervals, profile likelihood and sensitivity analysis are not bureaucratic extras; they prevent false certainty in scale-up decisions. Residual patterns are often more informative than a single fitted error statistic. Weighting can be necessary when measurement uncertainty grows with concentration.

The larger aim of Biokinetics is to answer this question: Construct and interrogate growth, inhibition, maintenance, and product-rate models; discriminate identifiable parameters from fitted numbers. A result must be interpreted as a statement about a specified physical boundary and observed quantities. Distinguish the underlying biological mechanism from the mathematical convenience that makes a calculation tractable. A model can be dimensionally consistent and still be badly chosen for the organism, medium or scale. Conversely, a complicated simulator is not superior unless its extra parameters can be identified and tested with observations.

A useful model and its assumptions

Working relation: For y=f(θ,x), local sensitivity is ∂y/∂θ. In the Monod model, ∂μ/∂μmax=S/(Ks+S), and ∂μ/∂Ks=−μmax S/(Ks+S)².

The displayed relationship is a starting point for reasoning. Before substituting numbers, name every term, give the unit and decide whether each quantity is a measured value, a fitted coefficient, or a controlled setting. Conservation laws are independent of constitutive assumptions, while kinetic and transfer coefficients depend on conditions. Make a sketch of the relevant system boundary and show what enters, accumulates and leaves. State whether the rate is per unit liquid volume, per unit catalyst or per unit cell mass; confusing these bases can lead to order-of-magnitude errors even when individual values seem realistic.

When a relationship contains an exponential, check which variable is integrated in the exponent and whether the exponent is dimensionless. When it contains a ratio, check what happens as the denominator approaches a limiting condition. When it describes an observable signal, identify the calibration that connects that signal to the claimed physical quantity. These checks are not substitutes for experiment, but they reject many wrong models before fitting begins.

Worked investigation with synthetic numbers

At S=20 g/L with Ks=0.2 g/L, S/(Ks+S)≈0.990. The rate is nearly μmax, so that experiment scarcely constrains Ks. At S=0.2 g/L, the model has a much stronger relative dependence on Ks.

Reconstruction task. Rewrite the stated computation as a ledger: input variable and unit, conversion used, intermediate result and final quantity. Show the same calculation with one input perturbed by 10% while retaining the others, and determine whether the output response is approximately proportional, nonlinear, or sensitive to a constraint. Then reverse the question: what input accuracy would be needed to distinguish the two predictions using a real instrument? Avoid claiming more significant figures than the chosen coefficients justify.

The example is intentionally synthetic. It should teach the direction and scale of a response, not imply that a specific organism or process will exhibit these numerical coefficients. If the result suggests substantial accumulation, ask where the required mass and energy originate. If it suggests extreme sensitivity, inspect whether the sensitivity arises from a genuine mechanistic feature or from mathematically approaching a singular assumption.

Measurement design and engineering judgment

Design a challenge experiment that can fail the model. Use concentrations spanning low and high substrate response, include controls, and test whether the fitted function predicts a held-out condition. Record pH, temperature and the assay method as part of the model definition.

Plan measurements around the saturation transition and replicate near the detection limit. Split calibration and challenge datasets by experimental run rather than random timestamp.

A defensible evidence plan identifies a process objective, measured signals, their frequency, instrument or assay uncertainty, and a comparison criterion selected before viewing the data. Include control conditions and independent replicates when the biological material varies across runs. Distinguish calibration measurements from hold-out verification data. If the main parameter is estimated from one narrow operating window, do not assume it will remain identifiable when oxygen, temperature, shear, pH or medium composition changes.

In a design review, explain the alternatives rather than selecting a model because it yields the highest projected titer or lowest cost. A common problem is conflating process capacity with process utilization: an available transfer rate is not automatically used, and biological demand can be low because the culture is inhibited. When a prediction is contradicted, investigate both the experiment and the model, and keep a record of the resulting revision.

Where the simple model fails

An optimizer that converges to a small residual does not prove a unique parameter set or correct model structure. Correlation among parameters can produce unstable forecasts.

A classroom model is especially fragile when extrapolated beyond its parameter-identification range. Carefully distinguish structural uncertainty (missing mechanism or wrong equation), parameter uncertainty (imprecisely estimated coefficient), and measurement uncertainty (sensor or assay error). Running many simulations with one incorrect equation does not resolve structural uncertainty. A useful sensitivity chart should therefore accompany a stated model boundary, not pretend to replace it.

For an independent verification, perform three checks: units and conservation; a limiting condition with a known qualitative answer; and a second calculation method or external measurement. Explain what each check can and cannot demonstrate. Automated agreement with an equation confirms the implementation, not necessarily the equation's suitability for a real reactor.

Guided laboratory and discussion

Open [Laboratory 06](../labs/lab-06.html). Record the default input values with units. Move the first slider to three equally spaced settings while leaving other controls unchanged; save the response to CSV and compare the graph with the governing relation in that laboratory. Then change a second independent parameter to test whether the original trend is stable. The laboratory describes its physical limits and reports an explicit warning where assumptions fail. The export is generated locally; no data leave the browser.

Investigative prompt: Propose five substrate levels that would best distinguish a Monod curve from an inhibited-growth curve within a defined safe experimental envelope.

Interpretation prompt: Identify a response that the model cannot predict, and propose one targeted measurement needed to decide if the omission matters. Write a two-sentence decision: what is supported by the simulation and what remains unproven. A high-grade response distinguishes a mathematical identity, an observation and a contingent engineering inference.

Lesson check

1. State the modeled system boundary, the type of measurable quantity being predicted and the appropriate units.
2. Derive or reconstruct the stated relation from a rate law or conservation balance, and show a limiting case.
3. Reproduce the worked calculation, then vary one input and interpret sensitivity without generalizing outside the stated model.
4. Identify two independent measurements that could falsify the simplified hypothesis.
5. Present a recommendation that is explicitly conditional on the evidence rather than an unsupported production specification.

Continue: Unit assessment. Return to the [unit learning pathway](../units/unit-02.html) or attempt the [unit assessment](../assessments/unit-02.html).

Unit 02 — Applied engineering project: Biokinetics

Status: fictional teaching case; all experimental measurements, process values and costs must be supplied or explicitly marked synthetic. This is not an authorization to conduct biological culture, run equipment, release a therapeutic or discharge wastewater.

Project brief

A feed-substrate concentration series gives an apparent Monod curve at low concentration and depressed growth at the high end. A proposed controller would keep substrate high to avoid starvation, while an alternative is a lower concentration fed-batch policy.

The task is not to maximize a single simulated variable. It is to demonstrate that an engineering conclusion rests on a defensible boundary, conserved quantities, experimentally testable assumptions, and explicit uncertainty. Students must distinguish a model output from a measurement and a possible risk from an observed failure. A written recommendation must name the conditions that would reverse it.

Analytical scope

Fit a Monod and inhibited-growth curve to synthetic observations that you create transparently with noise. Compare residual structure, parameter identifiability and held-out prediction. Calculate maintenance demand as a separate term and explain its effect on apparent biomass yield.

Construct a process map with numbered stream or information flows. Give a name, unit, origin, expected uncertainty, and timing for every requested input. Show where the model has a fitted coefficient as opposed to a measured quantity. If a coefficient is not available, express results parametrically and give an illustrative sensitivity range; do not silently invent a best-estimate industrial value.

Evaluate at least two candidate decisions with one common denominator: product mass per time, target recovery, demand/supply margin, analytical comparability, or another defensible quantity aligned with the case objective. A design that seems superior only because different quantities are being compared should be rejected as unresolved. Use a conservation audit to check whether any claimed improvement moves beyond the available mass, substrate or energy.

Numerical investigations

1. Extract a first-principles relationship from one of the four unit lessons; identify dimensions and limiting cases.
2. Run [Lab 04](../labs/lab-04.html) and record the full model input vector. Export its parameter sweep and explain the default trend.
3. Run [Lab 05](../labs/lab-05.html); deliberately choose a second configuration with different assumptions. Give the independent reason to use the second model.
4. Run [Lab 06](../labs/lab-06.html). Identify any warnings, invalid domains, or model-regime boundaries and explain their physical significance.
5. Use a spreadsheet, pseudocode or a short program to independently verify one selected model output. Report discrepancy, units and rounding conventions. Test one zero-input, saturation, washout or conservation limit appropriate to the chosen model.

Measurement and evidence plan

To challenge your preferred conclusion, describe the specific experiment, sample or instrument measurement that would cause the most information gain. Plan a negative or reference control, repeat measurements where biological variability matters, and specify the comparison that would be judged before seeing the results. Report whether the experiment challenges parameter values, constitutive form or sensor calibration. Maintain separate pages for evidence actually collected, assumptions borrowed from literature and hypothetical inputs introduced for illustration.

Deliverables

Submit source data, fitting script or spreadsheet, parameter units, confidence or scenario intervals, comparison plots, and a feeding policy conditional on independent tests of toxicity and oxygen transfer.

The final package should contain a concise executive summary, a source-and-assumption register, dimensional and material balances, transparent calculations, a three-scenario sensitivity chart, and the source CSV files needed to reproduce every figure. Include two alternatives not selected and explain why they fail against the stated objective or evidence requirements.

Quality rubric (100 marks)

Reflection: Which additional observation, if obtained tomorrow, would have the highest chance of changing your recommendation? Explain why this information is more valuable than running another deterministic curve with the same uncertain assumptions.

Unit 02 — Assessment: Biokinetics

This assessment has 16 substantive multi-part problems: four for each lesson. The purpose is to test domain reasoning, consistent units, conserved quantities, uncertainty and evidence. Students may explicitly reject an ill-posed process claim. Model values and numeric scenarios in the questions are fictional, not real plant operating conditions.

Assessment rubric (10 marks per task): clear physical boundary and correct units (2); correct equation or causal mechanism (2); numerical reconstruction or falsifiable reasoning (2); explicit applicability limits and uncertainty (2); defensible experimental or process decision (2). A correct calculation that silently uses an unsupported model cannot earn full credit.

Lesson 1 — Monod growth and nutrient limitation

Q01 — Reproduce and perturb a calculation (10 marks)

A. Reconstruct the numbers. Begin with the following numerical evidence: For μmax=0.5 h⁻¹, Ks=0.8 g/L and S=1.2 g/L, μ=0.3 h⁻¹. A batch with X=0.4 g/L and yield 0.5 g/g initially consumes 0.24 g substrate/L/h. That instantaneous rate cannot be maintained without checking the declining S. Identify every supplied number, every resulting number and the unit basis for each. Recompute the result using your own calculator or code, not by copying the printed final value. Alter one input by 15%, while preserving all others, and quantify the absolute and percentage change in output. Is the local response linear, and would a 15% observation error have the same impact as a deliberate 15% process change? State one limiting case and judge whether the scenario remains physically possible under conserved mass, energy or material balances.

Q02 — Conservation and dimensional audit (10 marks)

B. Develop the relation, not just the answer. Take the stated working relationship: μ(S)=μmax S/(Ks+S), dX/dt=μX and dS/dt=−μX/Yxs when maintenance is omitted. Biomass yield Yxs has units g biomass/g substrate. Begin from a species inventory, rate law or causal diagram appropriate to Monod growth and nutrient limitation. Show which terms would appear in the full statement, and identify each term removed in obtaining the simplified equation. Specify the units of two independent variables and of the predicted observable. Test at least two limiting conditions, including a zero rate, a dilute regime, a saturation boundary or a zero residence time where applicable. Explain how you would detect a wrong sign or a missing conversion factor without knowing the expected answer.

Q03 — Challenge the numerical model (10 marks)

C. Attempt to falsify the simulation. Open [Laboratory 04](../labs/lab-04.html) and record the exact default inputs with units. Save the full curve as CSV, choose three separated points, and calculate a finite-difference sensitivity from each adjacent pair. Repeat after changing a second parameter by a controlled amount. Use the model's stated restrictions—The model omits substrate inhibition, maintenance demand, cell death and transport heterogeneity. A fitted Ks may absorb oxygen limitation rather than represent nutrient affinity.—to identify at least two reasons a real experiment could disagree. Propose an observable and an experimental control that would discriminate parameter error from a missing physical mechanism. Do not equate agreement with the software to biological validation.

Q04 — Design a discriminating experiment (10 marks)

D. Make an evidence-constrained decision. Consider this engineering action: Estimate μmax and Ks jointly with confidence intervals across low and high substrate conditions. Inspect whether growth-rate observations at small S exceed the measurement error. Develop two plausible alternatives and a comparison matrix with expected benefit, scientific risk, measurement effort and an uncertainty or scale-up limitation. Propose three measurements that could change your decision, including one that directly tests the simplified model and one that checks product quality or conservation. Explain what evidence would make you stop the trial. Write a clear recommendation conditional on that evidence; explicitly say why a purely synthetic numerical prediction does not permit an operational or regulatory claim. End with a one-paragraph communication suitable for a multidisciplinary design review.

Lesson 2 — Substrate inhibition, maintenance and death

Q05 — Reproduce and perturb a calculation (10 marks)

A. Reconstruct the numbers. Begin with the following numerical evidence: Let μmax=0.6 h⁻¹, Ks=0.4 g/L, Ki=20 g/L. At S=2 g/L, μ≈0.462 h⁻¹; at S=20 g/L, μ≈0.297 h⁻¹. More food reduces predicted growth under the model. Adding mS=0.03 g/g/h raises the uptake prediction even if μ stays unchanged. Identify every supplied number, every resulting number and the unit basis for each. Recompute the result using your own calculator or code, not by copying the printed final value. Alter one input by 15%, while preserving all others, and quantify the absolute and percentage change in output. Is the local response linear, and would a 15% observation error have the same impact as a deliberate 15% process change? State one limiting case and judge whether the scenario remains physically possible under conserved mass, energy or material balances.

Q06 — Conservation and dimensional audit (10 marks)

B. Develop the relation, not just the answer. Take the stated working relationship: A common empirical form is μ=μmax S/(Ks+S+S²/Ki). A maintenance-inclusive uptake model is qS=μ/Yxs+mS; units of qS and mS are g substrate per g biomass per hour. Begin from a species inventory, rate law or causal diagram appropriate to Substrate inhibition, maintenance and death. Show which terms would appear in the full statement, and identify each term removed in obtaining the simplified equation. Specify the units of two independent variables and of the predicted observable. Test at least two limiting conditions, including a zero rate, a dilute regime, a saturation boundary or a zero residence time where applicable. Explain how you would detect a wrong sign or a missing conversion factor without knowing the expected answer.

Q07 — Challenge the numerical model (10 marks)

C. Attempt to falsify the simulation. Open [Laboratory 05](../labs/lab-05.html) and record the exact default inputs with units. Save the full curve as CSV, choose three separated points, and calculate a finite-difference sensitivity from each adjacent pair. Repeat after changing a second parameter by a controlled amount. Use the model's stated restrictions—A quadratic inhibition denominator is empirical and may fail outside its calibration range. The same decline in observed viable cells can arise from death or reduced specific growth.—to identify at least two reasons a real experiment could disagree. Propose an observable and an experimental control that would discriminate parameter error from a missing physical mechanism. Do not equate agreement with the software to biological validation.

Q08 — Design a discriminating experiment (10 marks)

D. Make an evidence-constrained decision. Consider this engineering action: Collect high-concentration observations rather than fitting inhibition using low-S data only. Compare alternative hypotheses: osmolarity, dissolved oxygen and assay bias. Develop two plausible alternatives and a comparison matrix with expected benefit, scientific risk, measurement effort and an uncertainty or scale-up limitation. Propose three measurements that could change your decision, including one that directly tests the simplified model and one that checks product quality or conservation. Explain what evidence would make you stop the trial. Write a clear recommendation conditional on that evidence; explicitly say why a purely synthetic numerical prediction does not permit an operational or regulatory claim. End with a one-paragraph communication suitable for a multidisciplinary design review.

Lesson 3 — Product kinetics and non-growth-associated synthesis

Q09 — Reproduce and perturb a calculation (10 marks)

A. Reconstruct the numbers. Begin with the following numerical evidence: With X₀=1 g/L, μ=0.12 h⁻¹ over 6 h, α=0.2 g/g and β=0.015 g/g/h, X≈2.05 g/L. Growth-associated product is ≈0.211 g/L, while the β contribution requires integrating X over time; using X₀ alone would underestimate it. Identify every supplied number, every resulting number and the unit basis for each. Recompute the result using your own calculator or code, not by copying the printed final value. Alter one input by 15%, while preserving all others, and quantify the absolute and percentage change in output. Is the local response linear, and would a 15% observation error have the same impact as a deliberate 15% process change? State one limiting case and judge whether the scenario remains physically possible under conserved mass, energy or material balances.

Q10 — Conservation and dimensional audit (10 marks)

B. Develop the relation, not just the answer. Take the stated working relationship: dP/dt = α dX/dt + β X is a simplified growth-associated plus non-growth-associated production relation for a constant-volume culture without product loss. α is g product/g biomass; β is g product/(g biomass·h). Begin from a species inventory, rate law or causal diagram appropriate to Product kinetics and non-growth-associated synthesis. Show which terms would appear in the full statement, and identify each term removed in obtaining the simplified equation. Specify the units of two independent variables and of the predicted observable. Test at least two limiting conditions, including a zero rate, a dilute regime, a saturation boundary or a zero residence time where applicable. Explain how you would detect a wrong sign or a missing conversion factor without knowing the expected answer.

Q11 — Challenge the numerical model (10 marks)

C. Attempt to falsify the simulation. Open [Laboratory 06](../labs/lab-06.html) and record the exact default inputs with units. Save the full curve as CSV, choose three separated points, and calculate a finite-difference sensitivity from each adjacent pair. Repeat after changing a second parameter by a controlled amount. Use the model's stated restrictions—The law is an empirical approximation and does not reveal intracellular regulatory mechanisms. With fed-batch dilution, write mass balances on V·P rather than on P alone.—to identify at least two reasons a real experiment could disagree. Propose an observable and an experimental control that would discriminate parameter error from a missing physical mechanism. Do not equate agreement with the software to biological validation.

Q12 — Design a discriminating experiment (10 marks)

D. Make an evidence-constrained decision. Consider this engineering action: Fit product and biomass observations jointly, then test the result on a held-out portion of the batch. Report product assay detection range and whether the product degrades. Develop two plausible alternatives and a comparison matrix with expected benefit, scientific risk, measurement effort and an uncertainty or scale-up limitation. Propose three measurements that could change your decision, including one that directly tests the simplified model and one that checks product quality or conservation. Explain what evidence would make you stop the trial. Write a clear recommendation conditional on that evidence; explicitly say why a purely synthetic numerical prediction does not permit an operational or regulatory claim. End with a one-paragraph communication suitable for a multidisciplinary design review.

Lesson 4 — Kinetic fitting, identifiability and extrapolation

Q13 — Reproduce and perturb a calculation (10 marks)

A. Reconstruct the numbers. Begin with the following numerical evidence: At S=20 g/L with Ks=0.2 g/L, S/(Ks+S)≈0.990. The rate is nearly μmax, so that experiment scarcely constrains Ks. At S=0.2 g/L, the model has a much stronger relative dependence on Ks. Identify every supplied number, every resulting number and the unit basis for each. Recompute the result using your own calculator or code, not by copying the printed final value. Alter one input by 15%, while preserving all others, and quantify the absolute and percentage change in output. Is the local response linear, and would a 15% observation error have the same impact as a deliberate 15% process change? State one limiting case and judge whether the scenario remains physically possible under conserved mass, energy or material balances.

Q14 — Conservation and dimensional audit (10 marks)

B. Develop the relation, not just the answer. Take the stated working relationship: For y=f(θ,x), local sensitivity is ∂y/∂θ. In the Monod model, ∂μ/∂μmax=S/(Ks+S), and ∂μ/∂Ks=−μmax S/(Ks+S)². Begin from a species inventory, rate law or causal diagram appropriate to Kinetic fitting, identifiability and extrapolation. Show which terms would appear in the full statement, and identify each term removed in obtaining the simplified equation. Specify the units of two independent variables and of the predicted observable. Test at least two limiting conditions, including a zero rate, a dilute regime, a saturation boundary or a zero residence time where applicable. Explain how you would detect a wrong sign or a missing conversion factor without knowing the expected answer.

Q15 — Challenge the numerical model (10 marks)

C. Attempt to falsify the simulation. Open [Laboratory 06](../labs/lab-06.html) and record the exact default inputs with units. Save the full curve as CSV, choose three separated points, and calculate a finite-difference sensitivity from each adjacent pair. Repeat after changing a second parameter by a controlled amount. Use the model's stated restrictions—An optimizer that converges to a small residual does not prove a unique parameter set or correct model structure. Correlation among parameters can produce unstable forecasts.—to identify at least two reasons a real experiment could disagree. Propose an observable and an experimental control that would discriminate parameter error from a missing physical mechanism. Do not equate agreement with the software to biological validation.

Q16 — Design a discriminating experiment (10 marks)

D. Make an evidence-constrained decision. Consider this engineering action: Plan measurements around the saturation transition and replicate near the detection limit. Split calibration and challenge datasets by experimental run rather than random timestamp. Develop two plausible alternatives and a comparison matrix with expected benefit, scientific risk, measurement effort and an uncertainty or scale-up limitation. Propose three measurements that could change your decision, including one that directly tests the simplified model and one that checks product quality or conservation. Explain what evidence would make you stop the trial. Write a clear recommendation conditional on that evidence; explicitly say why a purely synthetic numerical prediction does not permit an operational or regulatory claim. End with a one-paragraph communication suitable for a multidisciplinary design review.

Practice task linked to this lesson: Propose five substrate levels that would best distinguish a Monod curve from an inhibited-growth curve within a defined safe experimental envelope.