Replication Arena.

A dish of loose cells (small dots) and a few parent bots. The bots swim around and push cells together into piles. A pile that grows big enough, is roughly round and is left alone for a few seconds hatches into a new bot, which joins in. Bots are colored by generation: the parents, their offspring, the offspring’s offspring and so on.

Status line

Simulated time, loose cells left, bots alive, the highest generation reached and the largest pile against the hatch size.

Generations chart

Bots of each generation over time. A chain of colors climbing the chart means replication is continuing.

Presets

2021 paper: spheres; Pac-Man; Pac-Man dynasty (offspring copy the parent's shape); Starved dish (too few cells); Flood (plenty of cells).

Click the dish

Drop a handful of fresh loose cells where you click.

Loose cells

How much raw material the dish starts with. Restarts the dish.

Parent bots

How many bots start. Restarts the dish.

Mouth

The opening of a C-shaped bot, from a sphere (0) to a wide Pac-Man mouth.

Stickiness

How strongly loose cells cling together once touching.

Hatch size

How many cells a pile needs before it can mature.

Speed

Simulation speed.

Offspring are spheres

On (as in the lab): every new bot is a sphere whatever its parent was. Off: offspring copy their parent's mouth.

Restart / Step

Start over, or advance half a simulated second.

Try this: run “2021 paper: spheres” and note the highest generation, then “Pac-Man” with the same raw material. Then compare “Starved dish” and “Flood” to see how much the supply of cells matters.

Kinematic self-replication

Living things usually reproduce by growth and division. In 2021, researchers at Tufts and the University of Vermont reported a different route in xenobots, small bodies made from frog stem cells: the bots swept loose cells in their dish into piles, and some piles matured into new swimming xenobots. Building a copy from surrounding material, rather than growing one, is called kinematic replication, an idea John von Neumann explored theoretically in the 1940s.

Spherical xenobots usually managed only one round before the piles got too small. An AI search for better shapes found a C shape, like Pac-Man, that gathered bigger piles and kept replication going for up to four rounds.

The model

The arena is a 2D version of that process, simplified to run in real time. Cells are particles with simple stickiness; bots swim at constant speed and turn at random; a pile hatches by rule (size, roundness, left untouched). The rule-based hatching stands in for days of development in the lab.

Heredity

In the lab, a pile rounds up into a sphere whatever shape its builder had, so nothing about the parent is inherited. The “Offspring are spheres” switch lets you turn that off as a thought experiment: with it off, a Pac-Man’s offspring are Pac-Men too. Biobot Tierra takes this question much further.

Space and time

The dish is the unit disk (dish radius 1). The simulation advances in real time: each displayed frame is one step of length dt (the frame’s duration, capped at 1/30 s), repeated once per unit of the speed setting. Random numbers are not seeded, so two runs with the same settings differ.

Loose cells

Cells are overdamped point particles of radius r = 0.005 (10,000–80,000 of them; 40,000 by default). Each step, a cell moves by the summed pair forces from cells within 2r plus Brownian noise:

The pair force is a linear repulsion 0.03 (1 − d/r) for overlapping cells (d < r), and for r ≤ d < 2r an attraction shaped as a tent that peaks at d = 1.5r with height 0.006 × stickiness (the 0–1 slider). Neighbors are found with a uniform grid. Cells leaving the dish are reflected back inside the rim.

Bots

• A bot is a rigid disk of radius 0.045 with an optional wedge cut out around its heading: the mouth, with a half-angle of 0–70°. Its outline is a signed distance function.
• Bots swim at a constant 0.12 dish radii per second. Their heading turns at a fixed rate drawn per bot (±0.1–0.4 rad/s, so bots swim in arcs) plus rotational diffusion (D_r = 0.4 rad²/s). Overlapping bots are pushed apart, and the rim reflects a bot’s heading.
• A cell that comes within its own radius of a bot’s surface is moved out along the distance function’s gradient, and stamped with that bot’s id and the time: its last toucher. That is all the sweeping there is. A mouth carries cells because the cells in its wedge are pushed along by the inner faces as the bot moves forward.
• Bots never tire or die.

Piles and hatching

• Every 0.5 simulated seconds, cell counts are binned on a 128 × 128 grid over the dish and smoothed with a 3 × 3 box filter. A grid cell is dense when its smoothed count is above 1.8 × the mean loose-cell density (clamped to 4–8 cells), and dense grid cells are flood-filled (4-connected) into piles.
• Each pile gets a mass (cell count), a center of mass and a compactness: its area divided by 2π times its mass-weighted spatial variance, which is 1 for a uniform disk and lower for elongated or ragged piles.
• A pile is a hatch candidate when it holds at least the hatch size (600 cells by default), its compactness is at least 0.65, and no bot is within its radius plus 0.01. Candidates are matched between scans by position (within 0.04). One that stays a candidate for 3 simulated seconds hatches: the cells within its radius are removed and a new bot appears at its center.
• The parent is the bot that last touched the largest share of the pile’s cells (a mass-weighted vote over the last-toucher stamps), and the offspring’s generation is the parent’s plus one. The offspring is a sphere, or, with the toggle off, has its parent’s mouth.

Calibration and checks

• The hatch size was chosen to reproduce the lab’s qualitative result. At hatch sizes of 150–250, spheres replicated as fast as Pac-Men; it was raised to 600, where the gap appeared. So the shape effect depends on that calibration.
• Single runs at the defaults (about 160 simulated seconds each) gave: spheres, 2 hatches and stalled at generation 1; Pac-Man, 11 hatches and generation 2 and climbing; the starved dish, 1 hatch; the flood, 18 hatches and generation 3.
• These are single, unseeded runs, so there are no error bars. The seeded, deterministic version of this arena used for statistics is Biobot Tierra, where the mouth’s advantage did not survive a calibration sweep checked on fresh seeds.

The arena reproduces the lab’s two main observations, qualitatively:

• Shape matters. C-shaped parents scoop cells into their mouths and leave bigger piles, so replication runs more generations than with spheres.
• Raw material matters. Replication stops when loose cells run out; a starved dish stalls even with Pac-Man parents, while a flooded one cascades several generations deep.

With heredity switched on, a Pac-Man dynasty keeps its shape advantage in every generation instead of losing it after the first, which is what inheritance would add. That is a property of the model, not something seen in xenobots.

• Cells are 2D dots and bots are rigid shapes; real tissue is 3D and soft.
• Real bots are pushed by cilia through fluid; here they swim at a set speed with random turns.
• Hatching is a rule standing in for days of development.
• Bots here never wear out; real xenobots run on stored energy and eventually stop.