Bacterial Growth

Bacterial colonies are a prime example for growing active matter, where systems are driven out of equilibrium by proliferation. This model is a simplified version of unpublished work by Yoav G. Pollack and Philip Bittihn; similar models can be found in literature. Here, a bacterium is modelled by two soft disk "nodes" linked by a spring, whose rest length grows with a constant growth rate. When it has reached its full extension, the cell divides into two daughter cells with the same orientation.

This example is a showcase of a complex continuous system. Agents will be splitting into more agents, thus having agent generation in continuous space. The model also uses advanced agent movement in continuous space, where a specialized "move_agent" function is created. Advanced plotting is also done, since each agent is a specialized shape. It is also available from the Models module as Models.growing_bacteria.

using Agents, LinearAlgebra

mutable struct SimpleCell <: AbstractAgent
    id::Int
    pos::NTuple{2,Float64}
    length::Float64
    orientation::Float64
    growthprog::Float64
    growthrate::Float64

    # node positions/forces
    p1::NTuple{2,Float64}
    p2::NTuple{2,Float64}
    f1::NTuple{2,Float64}
    f2::NTuple{2,Float64}
end

function SimpleCell(id, pos, l, φ, g, γ)
    a = SimpleCell(id, pos, l, φ, g, γ, (0.0, 0.0), (0.0, 0.0), (0.0, 0.0), (0.0, 0.0))
    update_nodes!(a)
    return a
end
Main.SimpleCell

In this model, the agents have to store their state in two redundant ways: the cell coordinates (position, length, orientation) are required for the equations of motion, while the positions of the disk-shaped nodes are necessary for calculating mechanical forces between cells. To transform from one set of coordinates to the other, we need to write a function

function update_nodes!(a::SimpleCell)
    offset = 0.5 * a.length .* unitvector(a.orientation)
    a.p1 = a.pos .+ offset
    a.p2 = a.pos .- offset
end

Some geometry convenience functions

unitvector(φ) = reverse(sincos(φ))
cross2D(a, b) = a[1] * b[2] - a[2] * b[1]

Stepping functions

function model_step!(model)
    for a in allagents(model)
        if a.growthprog ≥ 1
            # When a cell has matured, it divides into two daughter cells on the
            # positions of its nodes.
            add_agent!(a.p1, model, 0.0, a.orientation, 0.0, 0.1 * rand(model.rng) + 0.05)
            add_agent!(a.p2, model, 0.0, a.orientation, 0.0, 0.1 * rand(model.rng) + 0.05)
            kill_agent!(a, model)
        else
            # The rest lengh of the internal spring grows with time. This causes
            # the nodes to physically separate.
            uv = unitvector(a.orientation)
            internalforce = model.hardness * (a.length - a.growthprog) .* uv
            a.f1 = -1 .* internalforce
            a.f2 = internalforce
        end
    end
    # Bacteria can interact with more than on other cell at the same time, therefore,
    # we need to specify the option `:all` in `interacting_pairs`
    for (a1, a2) in interacting_pairs(model, 2.0, :all)
        interact!(a1, a2, model)
    end
end

Here we use a custom move_agent! function, because the agents have several moving parts. Notice that the first derivatives of all degrees of freedom is directly proportional to the force applied to them. This overdamped approximation is valid for small length scales, where viscous forces dominate over inertia.

function agent_step!(agent::SimpleCell, model::ABM)
    fsym, compression, torque = transform_forces(agent)
    new_pos = agent.pos .+ model.dt * model.mobility .* fsym
    move_agent!(agent, new_pos, model)
    agent.length += model.dt * model.mobility .* compression
    agent.orientation += model.dt * model.mobility .* torque
    agent.growthprog += model.dt * agent.growthrate
    update_nodes!(agent)
    return agent.pos
end

Helper functions

function interact!(a1::SimpleCell, a2::SimpleCell, model)
    n11 = noderepulsion(a1.p1, a2.p1, model)
    n12 = noderepulsion(a1.p1, a2.p2, model)
    n21 = noderepulsion(a1.p2, a2.p1, model)
    n22 = noderepulsion(a1.p2, a2.p2, model)
    a1.f1 = @. a1.f1 + (n11 + n12)
    a1.f2 = @. a1.f2 + (n21 + n22)
    a2.f1 = @. a2.f1 - (n11 + n21)
    a2.f2 = @. a2.f2 - (n12 + n22)
end

function noderepulsion(p1::NTuple{2,Float64}, p2::NTuple{2,Float64}, model::ABM)
    delta = p1 .- p2
    distance = norm(delta)
    if distance ≤ 1
        uv = delta ./ distance
        return (model.hardness * (1 - distance)) .* uv
    end
    return (0, 0)
end

function transform_forces(agent::SimpleCell)
    # symmetric forces (CM movement)
    fsym = agent.f1 .+ agent.f2
    # antisymmetric forces (compression, torque)
    fasym = agent.f1 .- agent.f2
    uv = unitvector(agent.orientation)
    compression = dot(uv, fasym)
    torque = 0.5 * cross2D(uv, fasym)
    return fsym, compression, torque
end

Animating bacterial growth

Okay, we can now initialize a model and see what it does.

space = ContinuousSpace((14, 9), 1.0; periodic = false)
model = ABM(
    SimpleCell,
    space,
    properties = Dict(:dt => 0.005, :hardness => 1e2, :mobility => 1.0),
    rng = MersenneTwister(1680)
)
AgentBasedModel with 0 agents of type SimpleCell
 space:  continuous space with 14×9 divisions
 scheduler: fastest
 properties: hardness, dt, mobility

Let's start with just two agents.

add_agent!((6.5, 4.0), model, 0.0, 0.3, 0.0, 0.1)
add_agent!((7.5, 4.0), model, 0.0, 0.0, 0.0, 0.1)

The model has several parameters, and some of them are of interest. We could e.g. define

adata = [:pos, :length, :orientation, :growthprog, :p1, :p2, :f1, :f2]

and then run! the model. But we'll animate the model directly.

Here we once again use the huge flexibility provided by plotabm to plot the bacteria cells. We define a function that creates a custom Shape based on the agent:

using InteractiveDynamics
using CairoMakie # choose plotting backend

function cassini_oval(agent)
    t = LinRange(0, 2π, 50)
    a = agent.growthprog
    b = 1
    m = @. 2 * sqrt((b^4 - a^4) + a^4 * cos(2 * t)^2) + 2 * a^2 * cos(2 * t)
    C = sqrt.(m / 2)

    x = C .* cos.(t)
    y = C .* sin.(t)

    uv = reverse(sincos(agent.orientation))
    θ = atan(uv[2], uv[1])
    R = [cos(θ) -sin(θ); sin(θ) cos(θ)]

    bacteria = R * permutedims([x y])
    coords = [Point2f0(x, y) for (x, y) in zip(bacteria[1, :], bacteria[2, :])]
    scale(Polygon(coords), 0.5)
end

set up some nice colors

bacteria_color(b) = CairoMakie.RGBf0(b.id * 3.14 % 1, 0.2, 0.2)

and proceed with the animation

abm_video(
    "bacteria.mp4", model, agent_step!, model_step!;
    am = cassini_oval, ac = bacteria_color,
    spf = 50, framerate = 30, frames = 200,
    title = "Growing bacteria"
)