2.1. A framework for agent-based modeling of structured cell populations

We consider interactions between agents to depend on the agent type and their internal states that evolve continuously over time. With the agent we associate an agent type , where , and a state vector , which is a realization of a Markov jump-diffusion process:

(1)

where is a deterministic drift vector corresponding to ordinary differential equation (ODE); is state-dependent Gaussian white noise corresponding to the stochastic differential equation part (SDE); and are Poisson process jumps with path-dependent intensity and increments that define the jump process of agent type .

We define an interaction channel c that takes input agents of types with internal states , which we denote as , and creates output agents of types with states :

(2)

where are the indices of agent types involved in the interaction. The rate function of the rule is computed using the internal states of the input agents . The states of the output agents are initialized probabilistically according to the transition kernel that defines the conditional probability of creating output agent states given the input agent states .

In simple terms, an interaction channel represents a reaction or rule, analogous to the definition of reactions in biochemical kinetics. However, here the agents’ internal states modulate both the interaction rate function and the transition kernel. Thus, mapping a channel to a set of reactions amounts to enumerating all possible agent states, potentially yielding an infinite number of reactions. This map is effectively expressed through the interaction channel analogous to rule-based frameworks [23,41,42]. For our purpose, we consider cells as agents whose populations comprise different cell types. For example, we can describe asymmetric stem cell division of a stem cell S₁ differentiating towards S₂, via the rule , which could depend on the cell cycle position of the mother cell. Similarly, we could consider cell-cell interaction via rules such as , where encode ligand-receptor interactions and encodes the resulting transcriptional changes.

2.2. AgentBasedModeling.jl facilitates model specification and integrates easily with deterministic and stochastic modeling tools

AgentBasedModeling.jl package allows for convenient model specification and simulation of such models. The first step (Fig 1 Step 1) in the specification concern the definition of interaction channels (Eq 2). These are implemented with the @interaction environment with the lines of code following the specification:

where are sampled from the transition kernel . The @channel line defines the rate function of the interaction and the interaction stoichiometry while the line defines the initializations of the output agents given the transition kernel B. The @connections line is used to indicate which traits of which agents correspond to the symbols used in the expressions. The tuple denotes that the value of vector corresponds to the state of the first input agent at time t. These rules are matched to the individual instances of agents in the simulation.

The second step (Fig 1, Step 2) in the specification is to define the internal state dynamics for all agent types (Eq 1). This is done as an ODE, SDE, jump process, or a combination of them using ModelingToolkit.jl [39] and Catalyst.jl [40], leveraging the existing Julia programming language ecosystem.

2.3. AgentBasedModeling.jl enables exact simulation of stochastic agent-based models

AgentBasedModeling.jl provides a stochastic simulation algorithm to exactly simulate any agent-based model. The outline of the algorithm is given in Algorithm 1 and is based on the first reaction method for simulating Markov jump processes [43]. To sample the next interaction time in a simulation time-interval , the algorithm computes the trajectories of agent states in that interval. This is dependent on the model given for the agent state dynamics and can, for example, involve solving a system of ODEs, SDEs, or sampling a trajectory of a jump-diffusion process for each agent in the population. We then construct a set of possible interaction instances between agents for each interaction channel. For each channel, the simulation algorithm samples and executes the instance with the fastest interaction time.

Algorithm 1. Stochastic simulator for the population models.

Require: Simulation time-span [T₀, T], lookup horizon .

 Let and and consider the time window .

 Let be the initial population of agents at time t defined by pairs of agent types and states at time t.

 For all agents in simulate the state trajectories in the window .

 Let C be a set of interaction channels. For each with input agent types construct the set of all without replacement combinations of agents in the population that match the input types.

 while do

  For all channels c let be the interaction rate for a combination of agents a at time and let denote the upper bound of an interaction rate for all . Utilize one the following algorithms: the first reaction method where for each use the thinning algorithm [44] with rate and bound to sample next interaction times and choose the time and input agents a corresponding to ; or apply the Extrande algorithm [45] with rate bound to sample an interaction time and the input agents a.

  Find the channel c with the least next interaction time and the corresponding combination of input agents .

  if then

   Given the set of input agents with states at time and output agent types , create output agents with the given types and states at time sampled from the kernel . Remove input agents from .

   Simulate the state trajectories of the output agents in the time interval , add the output agents to , and let .

  else

   Let , and simulate the state trajectories of all agents in in time window .

  end if

 end while

The package offers three algorithms for sampling the next interaction instance of a given channel, and different algorithms can be combined within a simulation model for each channel (Fig 1 Step 3). Simulation algorithms are specified within the @interaction keyword using the @sampler keyword. The first method FirstInteractionMethod() samples interaction times for each instance of an interaction using the thinning algorithm [44]. The second algorithm uses the Extrande method [45] to sample the next interaction time for each interaction channel and can be used by specifying ExtrandeMethod(...). An alternative but potentially unchecked approximation to these aforementioned methods is the GillespieMethod().

The algorithms FirstInteractionMethod(bfn, L) and ExtrandeMethod(bfn, L) require specifying a symbolic function bfn returning a constant upper bound in the time interval [t, t + L], where L is a user-specified lookup time horizon. For convenience, the bound can be computed automatically using the optional boundtype keyword, e.g., ExtrandeMethod(L; boundtype: = unknown), which can assume:increasing,:decreasing or:unknown values. In these cases, the interaction rate function is bounded by assuming monotonically increasing or decreasing functions of time, i.e., the bound in a time interval [t, t + L] is the interaction rate at time t + L or t, respectively. When the keyword argument is:unknown, which is the default setting, the algorithm computes the upper bound of the interaction rate by evaluating the rate at each timestep in the interval [t, t + L] produced by simulating the agent state dynamics and taking the maximum of the computed rates on the fly.

Although our simulation methods are routed in Markovian dynamics, state-dependent interaction rates generate effective time-dependent event rates, leading to non-exponential waiting-time distributions even when the full process remains Markovian in an appropriately extended state space. In AgentBasedModeling.jl, such dynamics can be represented by augmenting agents with internal state variables that modulate event propensities. This provides a natural framework for modelling history-dependent cellular events, such as size-dependent cell division or infection-age-dependent lysis events (see Section 3).

2.4. AgentBasedModeling.jl provides tailored simulation outputs

Agents and their states can be tracked at individual and population levels (Fig 1, Step 4). We distinguish (i) snapshot and (ii) trajectory simulation, (iii) interaction snapshots, and (iv) temporal interaction networks, which we will describe below. Biological data examples of snapshot measurements include cell size, growth, or gene expression obtained from smFISH, flow cytometry, single-cell RNA sequencing, or multi-omics technologies. Trajectories present common measurements in population dynamics such as the microbiome, cancer growth, infections, or drug responses. Interaction snapshots represent agent states sampled at interaction events, such as cell birth, division, or cell-cell interactions. Temporal interaction networks are the most complex output, enabling the tracking of cellular agents and states, such as those observed in time-lapse microscopy or lineage tracing [46]. The handling of these outputs in the package is described below.

A simulation is processed by specifying a model, a set of parameters, and an initial condition. The simplest output is a snapshot of the state of the population at the simulation endpoint, which we denote as a snapshot simulation:

which requires specification of a , initial conditions and parameters . Furthermore, the package can provide trajectory simulation representing a series of population snapshots at equidistant time points. These can represent counts of agents and types or snapshots across the agents’ states. For example, for an agent A with state s this can be obtained through using

where PopulationSnapshot(A) counts agents of type A and StateSnapshot(A, s) provides a snapshot of the agent state every timestep Δt.

Other outputs concern interaction snapshots that track states of the input and output agents of specified interactions when events occur. These can be accessed by selecting the states in the @interaction environment using @saveinstate and @saveoutstate Keywords:

Finally, the most complex output is a temporal interaction graph that represents a function-valued directed graph whose edges represent trajectories of agent states and vertices of sampled interactions between these agents:

As the tool is implemented as a package for the Julia language, we can utilize existing statistical and visualization packages for analyses. For example, visualisation of such a graph can be conveniently implemented using MetaGraphsNext.jl from the Graphs.jl ecosystem.

3.1. Stochastic gene expression is coupled to growth and division

We consider a stochastic agent-based model of gene expression coupled to cell growth (Fig 2a) using a combination of ModelingToolkit.jl, Catalyst.jl and AgentBasedModeling.jl (S1 Code). Such models explain concentration homeostasis [47–49] and account for extrinsic noise in cells [17,50,51]. We first consider the intracellular dynamics of cells whose state is given by cell size s, added size , and protein count p that are specified using the @abm_variables keyword. Cell size s increases deterministically with exponential growth rate and can be modeled by an ODE. Proteins p are expressed in geometrically distributed bursts m modeled by a Markov jump process. The @reaction_network keyword provided by Catalyst.jl combines these dynamics, which are then passed to AgentDynamics structure from AgentBasedModeling.jl as shown in Step 3.1.

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Fig 2. Stochastic gene expression is coupled to cell size in growing and dividing cells.

(a) Cartoon of the model involving bursty protein expression in growing and dividing cells with growth rate and division rate . Molecules are stochastically partitioned at cell division. (b) Lineage tree of growing cell population with protein levels superimposed. (c-d) Simulation of protein content and cell size for a lineage tree (shaded). A representative single cell tracked over several divisions is highlighted (red). (e-f) Simulated birth protein and size distributions at cell birth across lineage trees (red) are compared with the analytical computations (black).

https://doi.org/10.1371/journal.pcbi.1014366.g002

Step 1: Stochastic gene expression with cell division and growth

Cells divide via the adder mechanism, where a stochastic size increment is added between divisions, as observed in many cell types. In Step 3.1, we implement the interaction channel via the @interaction keyword as follows. First, we assign variables , s, p to the agent states via the @connections keyword. The interaction channel can be written as

(3)

(4)

with a size-dependent division rate

(5)

where denotes the cell size control (see S1 Code) and denotes variability in the placement of the division site [52]. The keyword @channel specifies the channel stoichiometry and the division rate. During cell division, cells are split, and proteins are partitioned equally between the two daughter cells. The transition kernel is sampled from beta and binomial distributions within the @transition keyword. It defines the probability of the two daughter cells inheriting sizes and protein counts and respectively, given the mother cell divides with size s and protein count p. We sample this interaction channel with the Extrande method specified via the @sampler keyword. For convenience, the optional keyword boundtype = :increasing computes the bound automatically, assuming the division rate and cell size are monotonically increasing with time. The keywords @saveinstate and @saveoutstate record the cell states of cell size s and protein levels p before and after division for analysis.

Step 2: Stochastic gene expression with cell division and growth

Finally, in Step 3, the interactions and state dynamics are composed into a simulation model with the AgentsModel function. The function simulate() performs the simulation with parameters passed by SimulationParameters().

Step 3: Stochastic gene expression with cell division and growth

The stochastic simulation performed by AgentBasedModeling.jl results in a lineage tree of cells (Fig 2b). Cell state trajectories display growing cell size and protein levels interrupted by stochastic cell division events (Fig 2c-d). We validated our implementation by computing protein distribution and size at cell birth analytically [47,52], which are in excellent agreement with our exact simulation algorithm (Fig 2e-f).

3.2. Cell-cell communication enables collective switching

Cell-cell communication regulates interactions between cells from microbial communities to complex multicellular organisms. Bacteria release small, diffusible molecules that enable density-dependent regulatory mechanisms, known as quorum sensing. The process coordinates collective behaviors such as biofilm formation and virulence [53]. Similarly, in multicellular organisms, differentiation is orchestrated by signaling pathways that guide cells toward specific lineages. In disease, dysregulated communication allows tumor cells to hijack normal differentiation cues and reprogram their environment [54,55]. While deterministic models are traditionally used to model these phenotypes [56], these systems lend themselves naturally to agent-based modeling [57,58].

We here explore a general model of communication that includes stochastic effects of cell-cell coupling present in small and finite populations (Fig 3a). The model involves cellular agents that expand exponentially in cell size s with rate and produce inducer molecules x whose levels feed back on their production via a Hill-type production rate f(x,s):

(6)

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Fig 3. Cell-cell signaling enables collective switching in an agent-based cell model with diffusible autoinducer.

(a) Illustration of the model involving import and export of an autoinducer molecule produced in cells growing with rate and dividing with rate . We assume competition via a turbidostat, maintaining a constant cell number. (b) Trajectories of extracellular and intracellular levels of a random cell traced backwards in time (population of 20 cells). (c) Distribution of extracellular signaling molecule is unimodal while intracellular autoinducer shows bimodality. (d) Trajectories of extracellular (orange) and intracellular levels (blue) show significant temporal correlations at low population size (5 cells). (e) The corresponding distributions of extra- and intracellular autoinducer are bimodal. (f) Lineage of a cell population undergoing cell division and dilution in a turbidostat. Superimposed autoinducer level shows synchronised population-level transitions between active and repressed cell states.

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The autoinducer is synthesized within cells and then secreted into the extracellular environment, and extracellular molecules y can then be sensed and taken up by surrounding cells

(7)

We further assume that cells grow in microchambers that support a defined population size through dilution at cell division, keeping the population size constant. Once a cell divides, a random cell from the surrounding population is removed to make space for the new daughter cell:

(8)

(9)

Cell division follows the stochastic adder rule and dilutes molecules via binomial partitioning, as before.

We simulated time courses following cells over many cell divisions in the turbidostat (S2 Code). Tracing a cell that was alive at the end of the simulation backward in time, we observed that the environment followed intracellular levels but was generally less variable because it integrates signals from cells (Fig 3b). The distributions of the extracellular autoinducer were unimodal, while the intracellular levels showed bimodal distributions (Fig 3c). This suggests that cells coexist in the inactive and induced states and that the environment follows the mean across cells. Surprisingly, we observed a significant increase in temporal correlations between the extra- and intracellular levels when we decreased the population size in our simulations to what would be appropriate for a mother machine (Fig 3d). Moreover, extracellular levels were bimodal similar to intracellular levels (Fig 3e). The simulations suggested that cells collectively occupy inactive or induced states that reflect low and high environmental inducer states (Fig 3f).

3.3. Lineage fate after bacteriophage infection is determined by stochastic phage burst kinetics

Bacteriophage is a paradigm for studying cellular decision-making, both at the level of individual cellular pathways and at the level of population dynamics [59]. While these pathways have been described in the manner of Gillespie using detailed stochastic reaction network models [59–62], their population dynamics has been explored using deterministic ODE models [63–66].

We here extend a recently experimentally validated model of infection-lysis dynamics during the lytic cycle of bacteriophage [64], incorporating multi-stage infection dynamics and Erlang-distributed lysis timing. Our stochastic model consists of discrete cell agents (Cell) that take up nutrients to grow and divide, and interact with phages dispersed in the environment (Env). The model explicitly accounts for stochasticity in intracellular phage replication, cell growth and division rates. The environment contains nutrients M and extracellular phages P and a cell’s state consists of the age , cell size s, intracellular nutrients N, and intracellular phages Q.

Cells and the environment interact through interaction channels that model phage infection, cell lysis, cell division, and nutrient uptake from the environment. Cells become infected with phages (possibly multiple times)

(10)

where the age is reset upon the first infection if Q > 0 and zero otherwise. Once a cell is infected, it lyses with an age- and infection-dependent rate

(11)

and releases a burst of phages into the environment. Lysis occurs after a random time following a Gamma distribution with a fixed mean and coefficient of variation measured experimentally [67,68]. This non-Markovian infection dynamics can be implemented through an infection-age dependent lysis rate. The corresponding rate function is

(12)

models non-exponential infection times. These could be obtained from multi-stage infection dynamics with Erlang-distributed lysis times [64–66,69], which is a special case of the Gamma distribution. Meanwhile, uninfected cells will be able to divide

(13)

(14)

and partition intracellular nutrients between the daughter cells, similar to the gene expression model in Section 3.1. Uptake is modeled by importing nutrients from the environment into the cell:

(15)

Intracellular nutrients are then consumed through metabolic processes fueling growth, but can also be directed towards replicating phages in infected cells:

(16)

where is the cellular growth rate following Monod’s equation.

We simulated the model starting from a single cell and a fixed number of phages in the environment (S4 Code). The cell population initially expanded, peaked, and then rapidly collapsed (Fig 4b), a pattern also observed in deterministic models [64]. Surprisingly, in our stochastic model, both the peak size and timing were highly variable (Fig 4c). Peak size was inversely related to large bursts of phages released into the environment by cell lysis events (Fig 4d). We found that the distribution of extinction times was bimodal, displaying subpopulations that experienced early and late collapse (Fig 4e). Early collapse correlated with phage bursts from a few infected cells (blue lines, Fig 4d). We therefore analyzed the burst distributions from individual lysis events in these populations that we tracked using the keyword @saveinstate (S3 Code). Burst distributions were much wider for the early-collapsing lineages, exhibiting an order-of-magnitude increase in the distribution mode. Such burst stochasticity is not accounted for in deterministic models but has been quantified experimentally [70]. Recent advances in single-cell microfluidics [71,72] provide a fine-grained picture of phage-replication dynamics, and the modeling framework presented here could facilitate integrating these new single-cell insights into agent-based models of population dynamics.

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Fig 4. Stochastic lysis kinetics induces bimodal extinction time distributions in an agent-based phage infection model.

(a) Model illustration of cellular agents that grow and divide after taking up nutrients from the environment. Phages infect cells and stochastically replicate inside cells in a nutrient-dependent manner, and lyse cells, releasing bursts of phages. (b) Lineage following cell division, nutrient uptake, infection and lysis across a micro-colony of cells. Line color represents intracellular phage levels. (c) Population trajectory following infection of a single cell. (d) Burst of phages following infection is stochastic. (e) Extinction times show bimodal distribution with populations divided in early and late extinction. (f) Phage burst distributions of early and late extinct populations.

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3.4. Discussion of findings and extensions

Our first example — modeling stochastic gene expression in growing and dividing cells — demonstrates how gene expression can be coupled to cell growth and division processes. Cells in this model grow exponentially and divide according to an “adder” cell size control mechanism, while proteins are synthesized in bursty, size-dependent reactions. We validated the exact simulations against analytical predictions for protein distributions at birth, finding direct agreement with our implementation. While we recovered known single-cell patterns, such as size-protein distributions, our model could easily be extended to include metabolism or gene regulation. Our simulation preserves trajectories, tracking individual cells over time, enabling other lineage-tree analyses similar to those in live-cell microscopy or lineage-tracing experiments.

An alternative to the model with added size-dependent division rate, is a multi-stage process that also accounts for the same adder behavior, and these models have been used to study the effect of gene replication on mRNA and protein expression [48,73–75]. We coupled a simple model of bursty gene expression to a multi-stage model of division and replication using AgentBasedModeling.jl (S4 Code and Section A in S1 Appendix). The multi-stage model reproduces the same cell size distributions, but it alters the gene expression distributions (Fig A in S1 Appendix). This is because the model lacks stochastic concentration homeostasis, which also prevents it from admitting simple analytical solutions [47,48].

The second case study — cell communication — illustrates how the framework can model intercellular signaling common to a broad range of microbial and mammalian systems. Cells exchange molecules with the environment, creating a shared pool that cells respond to in a stochastic, density-dependent manner via a positive feedback mechanism. Our simulation results reveal that feedback and stochasticity can give rise to complex, emergent behavior such as the coexistence of induced and repressed states at the single-cell and population level. Notably, at low population sizes, we observed that the temporal correlation between intra- and extracellular levels increased when cells switched synchronously between global on- and off-states. These transitions are driven by intrinsic noise, which is typically ignored in mean-field models.

Our third application – bacteriophage λ infection and lysis – further illustrates how stochastic intracellular processes can determine the population fate. As cells grow and divide, phages replicate stochastically within the host. Upon reaching a critical state, infected cells lyse, releasing a burst of phages that can infect other cells. We explicitly modeled lysis timing and burst variability, which are supported by experimental measurements but typically neglected or incorporated using the linear chain trick into deterministic models [64–66,69]. Simulations revealed that the timing and magnitude of early lysis bursts determine the population trajectory. We observed a bimodal distribution of extinction times: some populations collapsed early after large initial bursts, while others grew for longer before eventually collapsing. This heterogeneity is not captured by mean-field models, which predict extinction only over infinite-time horizons as they average over burst timings and lysis events. By contrast, our agent-based framework can trace stochastic events and cell histories to identify rare, high-impact lysis events that determine population fate.

We investigated whether bimodal extinction distributions are unique to phage infections or a general feature of stochastic infection models. We developed an agent-based SIR model with infection-age-dependent rates and population heterogeneity [76], including factors like incubation periods, age structure, and vaccination status (Section B in S1 Appendix, S5 Code, Fig Ba in S1 Appendix), which have previously been studied using partial differential equations [35]. Extinction occurred either early without outbreak or after the first or second wave due to low infection numbers, leading to multimodal extinction time distributions similar to what we observed in the phage model (Fig Bb in S1 Appendix). While multimodality persisted when ignoring population structure using Gillespie simulations [77], the age-structured model predicted a lower outbreak probability and delays in extinction times of outbreaks consistent with incubation effects (Fig Bc in S1 Appendix). These findings highlight the interplay of stochasticity and population structure underpinning infection dynamics.

Together, these case studies highlight the versatility of AgentBasedModeling.jl. Across very different biological scenarios—intracellular gene regulation, intercellular signaling, and host-pathogen interactions—our framework provides a unified formalism of structured population dynamics through evolving internal states and stochastic interactions.