Teaching objectives
Why do we age at such different rates? A mouse barely makes it to 3 years, while its naked mole-rat cousin is potentially immortal. This secondary-school lab invites students to explore the mathematics behind biological ageing: from survival curves to cellular automata, through feedback loops and oxidative stress.
What you'll learn
Students practise the mathematical modelling of real biological phenomena: calibrating parameters from data, reasoning about the stability of dynamic systems, and making decisions under resource constraints.
- Students act as detectives: they tune the parameters a and b of the Gompertz law until their survival curve matches that of a hidden species (the horse).
- As actuaries, they design a curve with two boundary conditions (S(60) = 80 %, S(80) = 20 %) and receive a proximity score at each checkpoint.
- As designers, they build a fictional species with a median lifespan of 50 years and a curve that drops below 10 % before age 70.
- They experiment with the fatal loop (damage–mitochondria feedback): they discover the point of no return and manipulate antioxidants and exercise to cross it.
- They compare each species' strategy in phase space: they identify which parameter distinguishes long-lived species from short-lived ones.
- They explore a cellular automaton of living tissue: they activate and deactivate lifestyle factors (sleep, sport, vegetables, vitamins) under a budget of 30 points, with the goal that at age 70 the tissue retains at least 70 % healthy cells.
Key mathematical ideas
- The Gompertz law S(t) = exp(−(a/b)(e^{bt} − 1)) models the survival curve of any species with just two parameters.
- a controls the baseline mortality rate (mortality at age 0) and b the rate of ageing (how fast mortality grows with age).
- The median lifespan is the age at which S(t) = 0.5; shifting it requires adjusting a and b simultaneously.
- The damage–mitochondria loop is a dynamic system with positive feedback: above a certain threshold damage becomes self-perpetuating; below it the system recovers (two attractors).
- The cellular automaton translates local rules (healthy/damaged/senescent cell) into emergent tissue behaviour; lifestyle factors modify each cell's transition probabilities.
Room-by-room contents
Room 1 · Why do we age?
Laboratory introduction: the central question about biological ageing is posed and the guiding thread is presented (Gompertz, the oxidative loop and cells).
Student tasks
- Read the introductory text and advance to begin the journey.
Room 2 · The Detective: identify the species
A mysterious survival curve appears on the graph. The student adjusts the parameters a and b of the Gompertz law (λ(t) = a·e^(b·t)) until their curve matches the hidden one, thereby discovering which species is concealed (the horse, median lifespan ~25 years).
Student tasks
- Move the a and b sliders to fit the Gompertz curve to the mystery curve.
- Discover which species it is when the overlap is good enough.
Room 3 · The Actuary: design a curve with constraints
The student acts as an insurance actuary: they must find a and b such that 80% of the population reaches age 60 and only 20% reaches age 80. Each control point gives a score of 0–100 based on precision; a total of 160 is needed to pass the room.
Student tasks
- Adjust parameters a and b to satisfy S(60) ≈ 0.80.
- Fine-tune until S(80) ≈ 0.20 as well and the total score reaches 160.
Room 4 · The Designer: median lifespan of 50 years
The challenge is reversed: create an imaginary species with a median lifespan of exactly 50 years (S(50) ≈ 50%) whose curve drops below 10% before age 70. The student must satisfy both conditions simultaneously.
Student tasks
- Find the combination of a and b that places the median at 50 years (±3 years tolerance).
- Verify that the curve reaches the 10% threshold before age 70.
Room 5 · The Fatal Loop: point of no return
The oxidative loop model is introduced (stress → damage → more stress). With fixed biological parameters (fb = 5.5, damage = 0.35), the system collapses without intervention. Only the antioxidant and exercise sliders can be moved to cross the stability threshold.
Student tasks
- Move the antioxidant and exercise sliders while observing the trajectory in phase space.
- Make the system cross the threshold so the tissue stabilises instead of collapsing.
Room 6 · Each species' strategy
Each animal (mouse, human, parrot, tortoise, etc.) has different oxidative loop parameters: the student selects each species and observes its trajectory in phase space, identifying which parameter distinguishes long-lived species from short-lived ones.
Student tasks
- Select at least three species and compare their trajectories in phase space.
- Identify which loop parameter separates long-lived species from short-lived ones.
Room 7 · Living tissue: lifestyle factors
A cellular automaton simulates tissue ageing cell by cell. The student activates and deactivates lifestyle factors (sleep, sport, diet, vitamins) and observes in real time how the proportion of healthy, senescent and dead cells changes.
Student tasks
- Try different combinations of factors and observe the effect on the tissue.
- Once sufficient exploration has been done, press Continue to advance.
Room 8 · Mission: keep the tissue healthy
Final room with a budget constraint: the student has 30 health points to distribute among sleep, sport, vegetables and vitamins. The goal is for the tissue to retain at least 70% healthy cells at age 70, bearing in mind that excess sleep or exercise also has a cost.
Student tasks
- Distribute the 30 budget points among the four factors without exceeding the limit.
- Achieve ≥ 70% healthy cells in the simulator upon reaching year 70.
Rooms to project
The most striking ones to show and discuss in class.