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A biological lattice-gas cellular automaton (BIO-LGCA) is a mathematical model for analysing collective behaviour in populations of interacting, proliferating and migrating biological cells or organisms. This model has been inspired by the FHP-LGCA model for incompressible fluid flow (WIKI LINK), assumes freely mobile agents and is based on a statistical description of the environment. The first BIO-LGCA was introduced as model for collective migration (Fig. [cluster-formation]). While individuals can not be distinguished, emergent behaviours at the population level can be extrapolated. The BIO-LGCA is a discrete lattice- and agent-based model and permits multi-scale analysis and efficient large-scale simulations .

Basic principles

As a cellular automaton, a BIO-LGCA is defined on a regular lattice, where the nodes of the lattice assume a certain number of discrete states . Evolution of node states is defined by a local rule and performed in discrete time steps. The rule specifies how the new state of a node is determined given the old states of the node and its neighbouring nodes.

As a lattice-gas, the state space of a BIO-LGCA is linked to the lattice geometry (Fig. [lgca-node]). Each node can be occupied by “biological agents”, e.g. biological cells or organisms. A node is structured into a fixed number of channels that limit the node capacity. Channels are attributed with velocities that predetermine the movement of residing agents to neigbouring nodes during the migration step. Thus, reordering agents within the node’s channels may mimic certain tactic behaviours. Agents move along the links and interact on the nodes of the lattice. This interaction can change the number of agents at individual nodes (birth/death processes) and might depend on the states in neighbouring nodes which allows to model collective effects. Accordingly, mass, energy and momentum are typically not conserved in BIO-LGCA models.

[cluster-formation]

fig: [cluster-formation]

fig: [lgca-node]

Applications

The BIO-LGCA idea has sparked models for migration and interaction of cells in cultures and developing organisms. The key modelling idea is to combine basic interaction rules (see section [basic-interactions]) in order to characterise specific cell interactions and migration types. If applied to biological cells, BIO-LGCA models do not conserve momentum as they model active cells in low Reynolds number environments. Moreover, cell motion may be influenced by the interaction of cells with components of their microenvironment through haptotaxis or differential adhesion, contact guidance, contact inhibition, and processes that involve cellular responses to signals that are propagated over larger distances (e.g. chemotaxis). BIO-LGCA models have also been used to study angiogenesis  , Turing pattern formation  , and various aspects of tumour dynamics  . In particular, simulation and analysis of appropriate BIO-LGCA models permits to characterise various cancer growth and invasion scenarios   (Fig. [BIO-LGCA- applications]).

fig:[BIO-LGCA-applications]

Definition

A BIO-LGCA is defined by a lattice ${\mathcal {L}}$, a state space ${\mathcal {E}}$, a neighbourhood ${\mathcal {N}}$ and a local rule-based dynamics.

Lattice

The regular lattice ${\mathcal {L}}\subset \mathbb {R} ^{d}$ consists of nodes ${\bf {r}}\in {\mathcal {L}}$ and defines the “spatial geometry” in which the automaton operates. Each lattice node ${\bf {r}}\in {\mathcal {L}}$ is connected to its $b$ nearest neighbours by unit vectors ${\bf {c_{i}}},i=1,\ldots ,b$, called velocity channels ($b$: coordination number). In addition, a variable number $a\in \mathbb {N} _{0}$ of rest channels (zero-velocity channels) ${\bf {c_{j}}}={\bf {0}},\;b<j\leq a+b,$ is allowed (Fig. [lgca-node]). The parameter $K=a+b$ defines the .

Neighbourhood

The set ${\mathcal {N}}:={\mathcal {N}}^{b}={\mathcal {N}}^{b}(0)=\{{\bf {c_{1}}},{\bf {c_{2}}},\ldots ,{\bf {c_{b}}}\}$ is the neighbourhood template and defines the nearest neighbours for $0\in {\mathcal {L}}$.

Then, ${\mathcal {N}}({\bf {r}}):={\mathcal {N}}^{b}({\bf {r}})={\mathcal {N}}^{b}+{\bf {r}},$ specifies the set of lattice nodes which influence the dynamics of the state at node ${\bf {r}}\in {\mathcal {L}}$. ,

State space

To each node ${\bf {r}}\in {\mathcal {L}}$, a state ${\bf {s}}=(s_{1},\ldots ,s_{K})\in {\mathcal {E}}=\{0,1\}^{K}$ is associated which has the following interpretation: the Boolean variables $s_{j},j=1,\dots ,K$ are occupation numbers that indicate the presence ($s_{j}=1$) or absence ($s_{j}=0$) of an agent in the respective channel Failed to parse (unknown function "\cj"): {\displaystyle \cj} . This reflects an [def:exclusionp] which requires that not more than one agent can be at the same node within the same channel. As a consequence, each node ${\bf {r}}\in {\mathcal {L}}$ can host up to $K$ agents[def:kappa], which are distributed in different channels.

The ${\textrm {n}}({\bf {s}})$ and the ${\bf {J}}({\bf {s}})$ that correspond to a given state $s$ are denoted by ${\textrm {n}}({\bf {s}}):=\sum _{j=1}^{K}\,s_{j\,}\;,$ and

${\bf {J}}({\bf {s}}):=\sum _{j=1}^{b}\,s_{j\,}{\bf {c_{j}}}.$

Dynamics

Interaction and transport rule

The system evolution of the BIO-LGCA ${\boldsymbol {\eta }}(k)\rightarrow {\boldsymbol {\eta }}(k+1),$ where $k\in \mathbb {N}$ is the time parameter and Failed to parse (syntax error): {\displaystyle \boldsymbol\eta(k),\boldsymbol\eta}(k+1)\in\mathcal{E}^\mathcal{L}} are random variables specifying the lattice configurations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \boldsymbol\eta}(k)=\big(\boldsymbol\eta({\bf r},k)\big)_{{\bf r}\in\mathcal{L}},\quad\boldsymbol\eta}(k+1)=\big(\boldsymbol\eta({\bf r},k+1)\big)_{{\bf r}\in\mathcal{L}}}

at time $k$ and $k+1$, respectively, is defined by the subsequent application of a local stochastic interaction and a deterministic transport rule.

Interaction rule:

The interaction rule is a function Failed to parse (syntax error): {\displaystyle \mathcal{R}:\mathcal{E}\times\mathcal{E}^{\mathcal{N}}\rightarrow{\mathcal{E},} defined by ${\mathcal {R}}{\big (}{\bf {s}},{\bf {s_{\mathcal {N}}}}{\big )}={\bf {s^{\mathcal {R}}}}\quad {\mbox{with probability}}$ Failed to parse (syntax error): {\displaystyle W\big(({\bf s},{\bf s_{\mathcal{N}}})\rightarrow {\bf s^{\mathcal{R}}\big)}=W\big({\bf s}\rightarrow {\bf s^{\mathcal{R}}}| {\bf s_{\mathcal{N}}}\big)}, \quad {\bf s}, {\bf s^{\mathcal{R}}}\in \mathcal{E}, {\bf s_{\mathcal{N}}}\in \mathcal{E}^{\mathcal{N}},}

where the transition kernel $W$ has to satisfy the following conditions $W\;:\;{\mathcal {E}}\times {\mathcal {E}}^{\mathcal {N}}\times {\mathcal {E}}\rightarrow [0,1]\quad {\textrm {and}}\quad \sum _{{\bf {s^{'}}}\in {\mathcal {E}}}W\left(({\bf {s}},{\bf {s_{\mathcal {N}}}})\rightarrow {\bf {s^{'}}}\right)=1,\quad ({\bf {s}},{\bf {s_{\mathcal {N}}}})\in {\mathcal {E}}\times {\mathcal {E}}^{\mathcal {N}}.$ The interpretation is as follows: The local interaction rule ${\mathcal {R}}$ stochastically assigns a new state ${\bf {s^{\mathcal {R}}}}$ to a node depending on $({\bf {s}},{\bf {s_{\mathcal {N}}}})$, i.e. the current states of the node ${\bf {s}}$ and its nearest neighbours ${\bf {s_{\mathcal {N}}}}$.

Transport rule:

The transport rule $T:{\mathcal {E}}^{\mathcal {L}}\rightarrow {\mathcal {E}}^{\mathcal {L}}$ is defined by $T({\boldsymbol {\mu }})={\boldsymbol {\mu }}^{'},{\boldsymbol {\mu }}_{i}^{'}({\bf {r}})={\boldsymbol {\mu }}_{i}({\bf {r}}-m{\bf {c_{i}}}),$ where $i=1,\ldots ,K,{\bf {r}}\in {\mathcal {L}},\quad {\boldsymbol {\mu }},{\boldsymbol {\nu }}\in {\mathcal {E}}^{\mathcal {L}}.$ and $m\in \mathbb {N}$ is the ``agent speed``.

Microdynamical equation

The evolution of the BIO-LGCA is specified by the following microdynamical equation:

Failed to parse (unknown function "\substack"): {\displaystyle \begin{aligned} &&\eta_i({\bf r}+m{\bf c_i},k+1)\\ &:=&\sum_{\substack{{\bf s},{\bf s^{\mathcal{R}}}\in\mathcal{E}\\ {\bf s_{\mathcal{N}}}=({\bf s^{(1)}},\ldots,{\bf s^{(b)}})\in \mathcal{E}^{\mathcal{N}}, {\bf s^{(b)}}=(s_1^{(b)},\ldots, s_K^{(b)})}} s^{\mathcal{R}}_i\cdot\Big\{ \xi\big(({\bf s},{\bf s_{\mathcal{N}}}),{\bf s^{\mathcal{R}}}\big)\cdot\\ &&\hspace{1cm}\cdot \big[\prod_{j=1}^{K}\eta_j({\bf r},k)^{s_j}(1-\eta_j({\bf r},k))^{1-s_j}\big]\cdot\\ &&\hspace{1cm}\cdot \prod_{p=1}^{b}\cdot \prod_{j=1}^{K}(\eta_j({\bf r}-{\bf c_p},k))^{s_j^{(p)}}\big(1-\eta_j({\bf r}-{\bf c_p},k)\big)^{1-s_j^{(p)}} \Big\}, \end{aligned}}

where $\xi :{\mathcal {E}}\times {\mathcal {E}}^{\mathcal {N}}\times {\mathcal {E}}\rightarrow \{0,1\}$, is a Boolean random variable with Failed to parse (syntax error): {\displaystyle P(\xi\big({\bf s},{\bf s_{\mathcal{N}}},{\bf s^{\mathcal{R}}})=1\big)=W\big(({\bf s},{\bf s_{\mathcal{N}})\rightarrow {\bf s^{\mathcal{R}}\big),\quad {\bf s}, {\bf s^{\mathcal{R}}}\in \mathcal{E}, {\bf s_{\mathcal{N}}}\in \mathcal{E}^{\mathcal{N}},} [rvariable]

The interpretation of the microdynamical equation is as follows: the probability distribution of the random variable $\xi :{\mathcal {E}}\times {\mathcal {E}}^{\mathcal {N}}\times {\mathcal {E}}\rightarrow \{0,1\}$, is defined by the local interaction rule. At each time step, first, the interaction rule is applied at each node of the lattice. This is followed by a deterministic transport step (Fig. [lgca-propagation]): The concatenation of the rules leads to a new state ${\boldsymbol {\eta }}({\bf {r}},k+1)\in {\mathcal {E}}$ of a node ${\bf {r}}\in {\mathcal {L}}$ given the states ${\boldsymbol {\eta }}({\bf {r}}+{\bf {c_{p}}},k)\in {\mathcal {E}}$ of the nodes in the neighbourhood ${{\mathcal {N}}({\bf {r}})}$ of ${\bf {r}}\in {\mathcal {L}}$. In the transport step, an agent residing in channel $({\bf {r}},{\bf {c_{i}}})$ is moved to channel $({\bf {r}},{\bf {r}}+m{\bf {c_{i}}})$, where $m$ defines agent speed.

image image fig:[lgca-propagation]

Basic interactions

Basic interactions are modeled by specific choices of the probability distribution for the random variable $\xi$ $\xi :{\mathcal {E}}\times {\mathcal {E}}^{\mathcal {N}}\times {\mathcal {E}}\rightarrow \{0,1\}.$ Failed to parse (syntax error): {\displaystyle P(\xi\big({\bf s},{\bf s_{\mathcal{N}}},{\bf s^{\mathcal{R}}})=1\big)=W\big(({\bf s},{\bf s_{\mathcal{N}})\rightarrow {\bf s^{\mathcal{R}}\big),\quad {\bf s}, {\bf s^{\mathcal{R}}}\in \mathcal{E}, {\bf s_{\mathcal{N}}}\in \mathcal{E}^{\mathcal{N}}.}

In the following we present key examples.

fig:[lgca-basic-interactions-without-nbd]

Fluid dynamics

Lattice gas automaton models for incompressible fluid flow are based on collision rules conserving mass and momentum . These rules can be mimicked with a BIO-LGCA by the following probability distribution:

${\begin{aligned}&&P{\big (}{\bf {s}},{\bf {s_{\mathcal {N}}}},{\bf {s^{\mathcal {R}}}})=1{\big )}=W{\big (}({\bf {s}},{\bf {s_{\mathcal {N}}}})\rightarrow {\bf {s^{\mathcal {R}}}}{\big )}=W{\big (}{\bf {s}}\rightarrow {\bf {s^{\mathcal {R}}}}{\big )}\\&=&{\frac {1}{Z({\bf {s}})}}\cdot \delta {\big (}n({\bf {s}}),n({\bf {s^{\mathcal {R}}}}){\big )}\cdot \delta {\big (}{\bf {J}}({\bf {s}})-{\bf {J}}({\bf {s^{\mathcal {R}}}}),0{\big )},\\\end{aligned}}$

where Failed to parse (syntax error): {\displaystyle Z({\bf s})=\sum_{s^{'}\in\mathcal{E}} \delta\big(n({\bf s}), n({\bf s^{\mathcal{'}}})\big)\cdot\delta\big({\bf J}({\bf s})-{\bf J}({\bf s^{\mathcal{'}}}),0\big)\in \mathcal{E}}} is a normalisation factor; $\delta (x,y)={\begin{cases}1&{\mbox{if }}x=y\\0&{\mbox{else }}\end{cases}},$ where $x,y\in \mathbb {R}$ (Kronecker delta, WIKI link). This rule conserves agent number and momentum.

Random walk

Random walks are for example performed by cells such as bacteria and amoebae in the absence of any environmental cues. Random walk of agents can be modeled by random redistribution of agent velocities with the following probability distribution:

Failed to parse (unknown function "\begin{aligned}"): {\displaystyle \begin{aligned} &&P\big({\bf s},{\bf s_{\mathcal{N}}},{\bf s^{\mathcal{R}}})=1\big)=W\big(({\bf s},{\bf s_{\mathcal{N}})\rightarrow {\bf s^{\mathcal{R}}\big)=W\big({\bf s}\rightarrow {\bf s^{\mathcal{R}}}\big) \\ &=&\frac{1}{Z({\bf s})}\cdot\delta\big(n({\bf s}), n({\bf s^{\mathcal{R}}})\big), \\ \end{aligned}}

where $Z({\bf {s}})=\sum _{s^{'}}\delta {\big (}n({\bf {s}}),n({\bf {s^{\mathcal {'}}}}){\big )}=\left({\begin{array}{*{20}c}K\\n({\bf {s}})\\\end{array}}\right)$ is a normalisation factor. This rule conserves mass, i.e. agent number.

Birth process

Biological cells and organisms can proliferate. The BIO-LGCA concept allows to describe processes without mass conservation, e.g. birth of agents. This can be achieved with with the following probability distribution:

${\begin{aligned}&&P{\big (}{\bf {s}},{\bf {s_{\mathcal {N}}}},{\bf {s^{\mathcal {R}}}})=1{\big )}=W{\big (}({\bf {s}},{\bf {s_{\mathcal {N}}}})\rightarrow {\bf {s^{\mathcal {R}}}}{\big )}=W{\big (}{\bf {s}}\rightarrow {\bf {s^{\mathcal {R}}}}{\big )}\\&=&{\frac {1}{Z({\bf {s}})}}\cdot r\cdot {\frac {n({\bf {s}})}{K}}\cdot {\mathcal {H}}(n({\bf {s^{\mathcal {R}}}})-n({\bf {s}})),\end{aligned}}$

where ${\mathcal {H}}$ is the Heaviside function (WIKI link), $r\in \mathbb {R}$ is a birth rate, and

$Z({\bf {s}})=\sum _{s^{'}\in {\mathcal {E}}}r\cdot {\frac {n({\bf {s}})}{K}}\cdot {\mathcal {H}}(n({\bf {s^{\mathcal {R}}}})-n({\bf {s}}))$ is a normalisation factor. Extensions to birth/death processes are straightforward. Birth/death BIO-LGCA models bear close similarities to reactive lattice-gas cellular automata .

image [basic-with-nbhd-impact]

Alignment

Biological cells, e.g. fibroblasts, can align their velocities (Fig. [interaction-mechanisms]). Here, we introduce a probability distribution mimicking agent interactions which favour local velocity alignment and lead to collective motion or swarming behaviour.

${\begin{aligned}&&P_{\alpha }{\big (}{\bf {s}},{\bf {s_{\mathcal {N}}}},{\bf {s^{\mathcal {R}}}})=1{\big )}=W{\big (}{\bf {s}},\rightarrow {\bf {s^{\mathcal {R}}|{\bf {s_{\mathcal {N}}}},\beta }}{\big )}\\&=&{\frac {1}{Z({\bf {s_{\mathcal {N}}}},\alpha )}}\exp {\big (}\beta \,{\bf {D}}({\bf {s_{\mathcal {N}}}})\cdot {\bf {J}}({\bf {s^{\mathcal {R}}}}){\big )}\delta {\big (}n({\bf {s}}),n({\bf {s^{\mathcal {R}}}}){\big )},\end{aligned}}$

where

• ${\bf {s}},{\bf {s^{\mathcal {R}}}}\in {\mathcal {E}},{\bf {s_{\mathcal {N}}}}=({\bf {s^{1}}},\ldots ,{\bf {s^{b}}})\in {\mathcal {E}}^{\mathcal {N}}$,
• $\beta$ is an alignment sensitivity,
• Failed to parse (syntax error): {\displaystyle {\bf D}({\bf s_{\mathcal{N}})=\sum_{p=1}^{b} {\bf J}(s^p)} is the local director field, and
• $Z({\bf {s_{\mathcal {N}}}},\beta )=\sum _{s^{'}\in {\mathcal {E}}}\exp {\big (}\beta \,{\bf {D}}({\bf {s_{\mathcal {N}}}})\cdot {\bf {J}}({\bf {s^{'}}}){\big )}\delta {\big (}n({\bf {s}}),n({\bf {s^{'}}}){\big )}$ is a normalisation factor.

The interpretation is as follows: the argument of the exponential is maximised for parallel local momentum and director field. The particular choice of this probability distribution triggers agent alignment (cp. and Fig. [cluster-formation]).

Attraction/adhesion

Biological cells can interact via cell-cell adhesion (Fig. [interaction-mechanisms]). Agent attraction/adhesion can be mimicked with the following probability distribution.

${\begin{aligned}&&P_{\alpha }{\big (}{\bf {s}},{\bf {s_{\mathcal {N}}}},{\bf {s^{\mathcal {R}}}})=1{\big )}=W{\big (}{\bf {s}},\rightarrow {\bf {s^{\mathcal {R}}|{\bf {s_{\mathcal {N}}}},\beta }}{\big )}\\&=&{\frac {1}{Z({\bf {s_{\mathcal {N}}}},\alpha )}}\exp {\big (}\beta \,{\bf {G}}({\bf {s_{\mathcal {N}}}})\cdot {\bf {J}}({\bf {s^{\mathcal {R}}}}){\big )}\delta {\big (}n({\bf {s}}),n({\bf {s^{\mathcal {R}}}}){\big )},\end{aligned}}$

where

• ${\bf {s}},{\bf {s^{\mathcal {R}}}}\in {\mathcal {E}},{\bf {s_{\mathcal {N}}}}=({\bf {s^{1}}},\ldots ,{\bf {s^{b}}})\in {\mathcal {E}}^{\mathcal {N}}$,
• $\beta$ is an attraction sensitivity (adhesivity),
• ${\bf {G}}({\bf {s_{\mathcal {N}}}})=\sum _{p=1}^{b}n({\bf {s^{p}}}){\bf {c_{p}}}$ is the density gradient field, and
• $Z({\bf {s_{\mathcal {N}}}},\beta )=\sum _{s^{'}\in {\mathcal {E}}}\exp {\big (}\beta \,{\bf {G}}({\bf {s_{\mathcal {N}}}})\cdot {\bf {J}}({\bf {s^{'}}}){\big )}\delta {\big (}n({\bf {s}}),n({\bf {s^{'}}}){\big )}$ is a normalisation factor.

The interpretation is as follows: the argument of the exponential is maximized for parallel local momentum and density gradient field. The particular choice of this probabilty distribution favors agent attraction. A similar rule has been introduced in .

Multiscale analysis

BIO-LGCA models allow analysis of behaviours emerging at multiple temporal and spatial scales. In particular, one can distinguish microscopic and macroscopic scales, where the microscopic scale is much smaller than the typical agent size and is not explicitly considered in BIO-LGCA models. The macroscopic scale is much larger than agent size and refers to the behaviour of the agent population. A BIO-LGCA operates at a mesoscopic scale which is between the microscopic and the macroscopic scale: the mesoscopic scale coarse-grains microscopic properties but distinguishes individual agents. In many cases, the macroscopic behaviour of the mesoscopic BIO-LGCA can be analysed very well with a spatial mean-field description based on a partial differential equation . In particular, BIO-LGCA have been used for analysing properties at the macroscopic biological tissue level that result from local cellular interactions. Typical examples for observables at a macroscopic scale are cell density patterns and quantities related to the dynamics of moving cell fronts and cluster size distributions . Cell density patterns can often be assessed experimentally and provide, therefore, a means to relate BIO-LGCA model predictions to experimental observation.

Meanwhile, microscopic model descriptions in the form of stochastic differential equations have been derived from the mesoscopic BIO-LGCA formulation . BIO-LGCA interaction rules are typically chosen phenomenologically instead of being directly deduced from physical, microscopic single-cell interaction potentials. However, it is also possible to derive BIO-LGCA interaction rules from physically-motivated Langevin equations for migrating cells .

Extensions

The BIO-LGCA idea can be extended to multispecies models with different agent types where agent types may differ in their migration or interaction behaviours reflected in the specific interaction rule (cp. also . It also possible to extend the idea to heterogeneous populations, e.g. agents differ in their adhesivities .

Limitations

BIO-LGCA models are appropriate for low and moderate agent densities. For higher densities agent shape may matter and other models, e.g. the Cellular Potts model are good model choices (see for reviews of on and off-lattice models). It is also important to be aware of lattice artefacts, e.g. the checkerboard artefact (cp. ). In general, in two spatial dimensions the hexagonal lattice possesses less artefacts than the square lattice.

Simulator

include link to website:

https://imc.zih.tu-dresden.de//biolgca/