Guelph iGEM

Modelling

Introduction

Our goal was to understand and guide the design of our lead-responsive yeast. In our system, a riboswitch senses Pb²⁺ and permits translation of an integrase (Int2). Int2 flips a state-switchable promoter, committing the cell to a long-lived “ON” state that either glows (diagnostic) or expresses a transporter to help remove lead (functional). Even a short lead pulse can be “remembered” by the cells because the flip is effectively irreversible on experimental time scales. This makes the system useful for detecting lead exposure and for gradually scrubbing lead from water.

How modelling fits: we turned that biology into a small set of ordinary differential equations (ODEs) that track the key moving parts over time (lead outside the cells, the riboswitch mRNA, integrase, the fraction of cells that have flipped “ON,” and the chosen output reporter/transporter). With these equations we can (i) predict the signal we’ll see for a given lead pulse, (ii) identify which parameters most affect performance, and (iii) point the wet lab to the most valuable measurements next.

Micro-glossary

  • ODE: an equation describing how a quantity changes in time (e.g., “how fast does integrase rise/decay?”).
  • Rate law: the formula for a reaction rate as a function of species (e.g., Michaelis–Menten term L/(K+L)).
  • Parameter: a fixed constant in the model (e.g., a decay rate or a half-saturation constant).
  • State variable: a quantity the model updates over time (e.g., extracellular lead concentration).

How to Build an ODE Model

Goal: Start with a real-world process (biology, chemistry, ecology, etc.) and end with a ready-to-simulate set of differential equations.

1. Clarify the Question

What outcome do you care about?
Over what time scale?
Which factors are essential vs. nice-to-have?
Example: Predict how a yeast population removes lead from water over 48h.

2. Draw a concept map/box-and-arrow diagram

State variables → boxes (populations, concentrations, volumes, etc.)
Processes/fluxes → arrows (growth, decay, uptake, inflow, etc.)
This picture becomes your equations road map.

3. Choose the state variables y(t)

Each variable should be:

Quantifiable (measurable or estimable)
Dynamically changing over your timescale
Non-redundant (avoid two variables that always sum to a constant)
Good picks: Cell count, substrate mass, mRNA copies
Usually not a state variable: Rate constants, yields, “time”, model parameters

4. Write process rules for each arrow

Mass-action? aXY
Saturation? VmaxS/(KM + S)
Logistic limiation? rN(1 − N/K)
External forcing? inflow F(t)
Keep units consistent while you write them.

5. Form the balance equations

For every state variable:

dy dt = (inflows) − (outflows)

Check each term is in the same units as y/t.

6. List parameters and initial conditions Table of parameters and initial conditions
7. Non-dimensionalize or scale

Often simplifies the system or reveals “small” parameters that can be dropped.
Optional but at least check that dimensionless groups make intuitive sense.

8. Qualitative checks

Signs: does each term increase or decrease the right variable?
Edge cases: if y = 0, is dy
dt ≥ 0 unless you add more?
Units: every equation should balance.

Software Project – ODE Generator

Motivation: For iGEM teams, building models from scratch each time is slow and error-prone. We built a small “ODE generator” that turns a minimal biological description (species + reactions) into a working simulation. It starts with sensible defaults and derivations, while still letting users override anything (custom rate laws, parameters, even full custom ODEs) when they need precision.

  • User input: species and reactions
  • Defaults & derivations: assigns standard rate laws/initial guesses
  • ODE builder: constructs the symbolic model
  • Numerical solver: integrates the equations over time
  • Outputs: plots, CSV exports, and a short report

Why this matters. Early on we learned that relying only on fixed defaults can miss behaviour present in curated models. Supporting custom rate laws and parameters preserves ease-of-use without sacrificing fidelity, addressing our “minimal input vs accuracy” tension.

Technical Summary of iGEM Guelph’s Model

ODEs to represent iGEM Guelph’s biological system were developed using the procedure outlined in How to Build an ODE Model with support from Dr. Kolja Kypke.

System Being Modelled.

A lead-responsive transcriptional control module in yeast: Pb²⁺ toggles a riboswitch, enabling translation of Int2, which flips a DNA segment. Flipped cells express either GFP (diagnostic) or a Pb²⁺ transporter (functional).

State Variables.

We track the 6 state variables listed below.

Table 1: State variables used in the model.

Table of parameters and initial conditions
Differential Equations.

All time derivatives are with respect to culture time t (s).

Table of parameters and initial conditions
Parameter Ledger.

Table 2: Model parameters, provisional values, sources, and planned assays. LIT = determined from literature; EXP = requires experimental measurements.

Table of parameters and initial conditions

Notes. The model separates two practical strains:

Diagnostic: qmax = 0 (L is fixed by the environment). This case admits closed-form solutions for the early layers (R and I), simplifying parameter fitting.

Functional: qmax > 0 (cells actively lower L via the transporter). Here, L is dynamic; the system loses the closed-form tractability but captures the sequestration we care about in water cleanup.

Results & Insights

Lead pulses “write” memory. Because promoter flipping is effectively irreversible, the final flipped fraction F is a record of how far the integrase reaction progressed while lead was present. Practically, brief exposures can be detected later (diagnostic strain), and in the functional strain, the culture tends to relax to a partial-memory state as exterior lead is exhausted.

Three levers that matter most:

  • Lead pulse characteristics (height/duration) — set the window during which memory can be written.
  • Integrase stability (γI) — faster decay shrinks the writing window.
  • Sequestration capacity (qmax, KH, and downstream expression via F → gRNA → G) — controls how quickly the functional strain collapses extracellular lead.

Assumptions & Limitations

Assumptions:

  • Well-mixed culture, constant volume;
  • Deterministic ODEs (no intrinsic noise);
  • Parameter stationarity over each simulated interval;
  • Promoter flip is effectively irreversible on our time scales.

Limitations:

  • No stochasticity/spatial effects yet;
  • Default parameterization is only a starting point — accuracy benefits from curated/custom values;
  • Functional strain couples biology to the environment (L is dynamic), which removes some analytic tractability but is necessary to capture sequestration.

Future Directions

  • Model validation: use Wet Lab data; adjust parameters and structure as needed.
  • Stochastic/hybrid models: add SSA or hybrid ODE-SSA for promoter-flipping noise.
  • Parameter inference: automated fitting seeded by literature priors.
  • Larger networks: extend the generator to multi-module circuits.
  • Assay-to-solver loop: when a measurement lands, the solver auto-refreshes plots.
  • Reusable templates: a gallery other teams can fork (diagnostic vs functional patterns).

References

  1. Beisel, C. L., & Smolke, C. D. (2009). Design Principles for Riboswitch Function. PLoS Computational Biology, 5(4), e1000363. https://doi.org/10.1371/journal.pcbi.1000363
  2. Bonnet, J., Subsoontorn, P., & Endy, D. (2012). Rewritable digital data storage in live cells via engineered control of recombination directionality. PNAS, 109(23), 8884–8889. https://doi.org/10.1073/pnas.1202344109
  3. Jones, A. M., et al. (2024). Genome-wide mRNA half-lives in S. cerevisiae. Nucleic Acids Research, 52, e112.
  4. Wu, L., et al. (2023). Context-dependent degradation of GFP in yeast. Molecular Cell, 83, 1742–1755.