Many cancer cells make energy by fermenting sugar, even with plenty of oxygen around. That is the Warburg effect. Here, a small tumor grows inside healthy tissue, and every cell decides how to make its energy by solving a small linear algebra problem. Pick a question, press Play, and watch.
A cell can respire one glucose using oxygen in its mitochondria: slow to set up, but it makes about 32 ATP (energy units). Or it can ferment it: fast and cheap, but only 2 ATP, plus 2 lactate (acid) as waste.
Write the cell's chemistry as a matrix S: one row per molecule, one column per reaction. A list of reaction speeds v keeps every molecule in balance when Sv = 0. All the balanced choices form the null space of S, and it turns out to be built from exactly two pathways:
ferment = (1, 1, 0, 2, 0, 2)
respire = (1, 1, 2, 0, 2, 0)
So every cell's metabolism is a × ferment + b × respire: a glucose fermented, b respired.
A cell chooses a and b to make the most ATP within its limits: the glucose it can take in, the oxygen around it, and how much its mitochondria can handle. Biologists call this flux balance analysis. A cell with little oxygen, or small mitochondria, has to ferment.
Capillaries (dark dots) supply oxygen and glucose and wash acid away; everything spreads between squares. Each step, every cell eats and makes ATP. Then it dies if it has too little ATP or too much acid, divides into an empty neighboring square if it has plenty, or otherwise stays.
Healthy cells can only grow into empty space. If a tumor cell and a healthy cell both want the same empty square, the tumor wins 3 times out of 4: cancer cells ignore the signals that tell normal cells to stop dividing. Healthy cells can win: they take back space when the tumor dies, and with stronger acid tolerance (Advanced settings) they hold the tumor off.
Why this doesn't match real tumors. Real tumors keep fermenting, so the model is missing something. The leading ideas: tumors evolve resistance to acid (here tolerance is fixed); dividing cells need sugar's carbon as building material, not just energy (here cells only count ATP); tumors physically push into tissue; and blood supply inside tumors is patchy, leaving oxygen-starved cores. Any of these could tip evolution back toward fermenting, and they are good next experiments.
A teaching model: the budget, the cost of mitochondria and the acid tolerances are made-up numbers chosen so the effects are visible. ATP yields (2 and 32) are standard textbook values.
These tumor cells ferment almost all their sugar and give off lactate, an acid. Healthy cells die in strong acid; tumor cells can take three times more.
Watch for: a ring of empty space spreading out ahead of the tumor.
Try the other views: shows the acid front moving out ahead of the tumor. shows the cleared zone filling with oxygen: the tumor isn't short of oxygen; it wins with acid.
The tumor ferments about 97% of its sugar and gives off lactate. Wherever the acid passes 1, healthy cells die, opening empty squares that the tumor divides into. By around step 110 every healthy cell is gone. The tumor itself only survives near capillaries, where blood washes its acid away; elsewhere it keeps poisoning itself. This is the acid-mediated invasion idea from cancer research: a tumor clearing space with its own acid.
What it suggests: two ways to stop this kind of invasion are to neutralize the acid or to help healthy cells tolerate it (try healthy acid tolerance 2 in Advanced settings: the tumor stays contained). Researchers have tested the first idea: in mice, drinking baking soda (bicarbonate) made tumors less acidic and reduced how often they spread1. The effect was small and varied between tumor types, and it isn't an approved treatment, so it is a lead rather than a cure.
Each tumor cell splits a fixed budget between sugar pumps and mitochondria (the cell's respiration machinery). When a tumor cell divides, its daughter's split changes a little at random. Cells whose split works better leave more descendants.
Watch for: tumor color shifting from orange (fermenting) to purple (respiring), and the bottom chart.
Try the view, and run past step 100. While the tumor spreads, acid rises (more and more tumor cells). Once it has taken over, acid fades as the tumor turns purple: by step 300 each tumor cell makes about 5 times less acid than at the start. is lower inside the tumor than in empty space, because respiring cells use it.
Every tumor cell splits its budget between sugar pumps and mitochondria. Daughters inherit the split with a small random change. Cells with more mitochondria get more energy out of each sugar (sugar is scarce in crowded tissue) and make less acid, so they survive and divide more. Over about 150 steps the average share spent on mitochondria rises from 50% to about 90%, and the share of sugar fermented falls from about 0.9 to about 0.3 (bottom chart).
The early, fermenting tumor still did the damage: its acid cleared out the healthy tissue first. So in this model fermenting is a temporary weapon: useful for invading, then abandoned. The tumor also settles with its acid just under 3, the most it can survive.
The puzzle: real tumors keep fermenting, so the model is missing something (see "What it found, and its limits" above). What it suggests, as a question rather than a result: if acid mainly helps tumors while they invade, therapies aimed at tumor acid might matter most early on.
Here every tumor cell re-decides each step how to split its budget, always picking whatever makes the most energy right now. Fermenting is the fastest, so they all switch to it, without "knowing" that their acid builds up around them.
Watch for: what happens to the tumor in the first few steps (use Step).
Try the view and press Step a few times: a burst of acid appears where the tumor was, then the capillaries wash it away and healthy cells grow back in.
The tumor disappears in a single step. Each cell picks pure fermentation because, right now, it gives the most energy (40 ATP versus 35 for a mixed split). But pure fermentation dumps about 8 units of lactate per step onto the cell's own square. The acid there jumps above 5, past the tumor's limit of 3, and every tumor cell dies at once. Healthy cells then grow into the empty space.
Why it matters: the greedy rule only sees energy; it never "counts" the acid it makes. Evolution in question 2 does count it, because cells that poison themselves leave no descendants. A choice that looks best in the moment can be self-destructive.
What it suggests: a tumor's own acid is a weak point. Real cancer cells survive their fermentation by pumping lactate out through transporter proteins. Researchers are testing drugs that block one of these pumps (MCT1), so lactate builds up inside the cancer cell; one, AZD3965, was found tolerable in an early human trial2. Our model only tracks acid outside cells, so this is an analogy, not a prediction.
How many cells of each kind
How much the tumor ferments
Changes apply on the next step. Restart to begin again with them.
Evolution is set by the questions above.
Click any square on the map to see the linear algebra problem that cell is solving.
The cell as a point (a, b): a = sugar fermented, b = sugar respired, per step. Dashed lines are its limits; the shaded area is everything it could choose.