Kinetic Equation for Cell Migration

• The kinetic equation for cell migration is a mathematical framework that uses conservation laws and energy principles to model cell density dynamics with active migration and force balance.
• It incorporates mesoscopic velocity-jump models and non-local sensing to capture stochastic turning, biased migration, and adhesion-mediated interactions.
• The approach bridges cellular behavior and macroscopic PDEs, enabling insights into tissue morphogenesis, cancer invasion, and immune cell dynamics through multiscale modeling.

A kinetic equation for cell migration is a mesoscopic or macroscopic mathematical framework describing the statistical distribution and transport of cells as a function of position and usually velocity (and possibly other internal states) over time, incorporating both stochasticity at the cellular level and physical-chemical-biological stimuli from the microenvironment. Modern approaches integrate multiscale modeling, force and chemical balances, non-local environmental sensing, and biochemical activity, unifying cell-centered and population-level perspectives to account for the emergent spatial–temporal migration dynamics observed in developmental, immune, and cancer contexts.

1. Balance Laws and Energy Principles

The derivation of kinetic equations for cell migration often begins with conservation laws and constitutive relations that encode the physics and biophysical context. The rigorous construction in Tatone–Recrosi–Tomassetti (Tatone et al., 2024) uses three coupled balance laws:

• Species molar balance: Cell concentration $c(x,t)$ (mol/m$^3$) and flux $J(x,t)$ obey local conservation,

$\frac{\partial c}{\partial t} + \nabla\cdot J = 0.$

• Force and microforce balances: Extension of Gurtin’s microforce theory incorporates the internal microstresses $\xi$ and microforces $s$, coupling to variations in concentration gradients and providing a mechanism for energy-dissipating interactions in active, deformable cellular systems.
• Free energy and chemical potential: The Ginzburg–Landau energy

$$\psi(c, \nabla c) = f(c) + \frac{1}{2}k_g |\nabla c|^2$$

yields a chemical potential as a variational derivative,

$$\mu_{\mathrm{thermo}} = \frac{\partial f}{\partial c} - \nabla\cdot(k_g \nabla c),$$

extended by an active migration term $\mu_{\mathrm{active}}(c)$ to represent motility-driven “uphill” diffusion.

The dissipation inequality furnishes a generalized Fickian flux law,

$J = -M(c)\nabla\mu(c, \nabla c; \ell),$

where $M(c) > 0$ is a motility coefficient and $\ell$ encodes external vectorial (e.g., chemotactic/haptotactic) cues (Tatone et al., 2024).

2. Mesoscopic and Macroscopic Kinetic Models

Kinetic equations in this context track the probability density $f = f(t, x, v, \ldots)$ of finding a cell at position $x$ with velocity $v$ (and possibly internal state variables), subject to processes such as velocity jumps (tumbles), non-local environmental sensing, and cell–cell or cell–matrix interactions. The canonical velocity–jump equation takes the form

$$\frac{\partial f}{\partial t} + v\cdot\nabla_x f = \mathcal{J}[f],$$

where $\mathcal{J}$ implements stochastic turning, directional reorientations, and possibly state transitions (Zhigun et al., 2020, Conte et al., 2022, Lorenzi et al., 2024).

Biophysical mechanisms incorporated:

• Biased turning: Transition kernels include chemotactic, haptotactic, and contact guidance cues, often through non-local integrals that model the spatial heterogeneity of fibers, ECM, or chemical fields (Loy et al., 2020, Loy et al., 2019, Conte et al., 2022).
• Active forces and motility: Recent energetic and force-balance models encode the effect of actomyosin contractility, protrusions, and substrate mechanics either in the chemical potential driving migration or through explicit force terms (Winkler et al., 17 Feb 2025, Perthame et al., 9 Jan 2026).
• Cell–cell/cell–matrix adhesion: Non-local adhesion is represented by convolution integrals for the force field and encounter rates, sometimes coupled to subcellular adhesive state variables (Zhigun et al., 2023).

The reduction to closed macroscopic PDEs commonly proceeds by moments expansion or Chapman–Enskog/Hilbert procedures, producing drift–diffusion or more general advection–diffusion (and occasionally fourth-order, Cahn–Hilliard-type) equations for the cell density. This connects with classical models such as Keller–Segel chemotaxis but supports more general motility and phase separation phenomena (Zhigun et al., 2020, Tatone et al., 2024, Perthame et al., 9 Jan 2026).

3. Modeling Non-local Sensing, Environmental Structure, and Activity

Non-local Sensing: Cells probe their microenvironment using protrusions or molecular sensors, integrating external cues over spatial scales comparable to or exceeding their size. Kinetic models encode this using non-local integrals—e.g., the turning kernel samples ECM, fibers, or chemical fields along a ray up to a sensing radius, weighted by a sensitivity function $\gamma(\lambda)$ (Lorenzi et al., 2024, Loy et al., 2019, Loy et al., 2020). This leads to transition probabilities

$$T(x, v, \omega) = \int_0^{R(x, \omega)} \gamma(\lambda) Q(x + \lambda \omega, \ldots) d\lambda,$$

where $Q$ may depend on fibers, matrix, or chemoattractant.

Physical Constraints: Sensing radii may be truncated by obstacles, matrix density, or crowding, introducing anisotropic or direction-dependent migration limits (Loy et al., 2019, Conte et al., 2022). These effects control the transition between hyperbolic (advection-dominated) and parabolic (diffusion-dominated) macroscopic regimes.

Active Migration and Pattern Formation: Activity—motility-driven “uphill” diffusion—can destabilize uniform steady states, producing pattern formation via a spinodal regime when $\partial\mu_{\mathrm{active}}/\partial c < 0$, as formalized in active Cahn–Hilliard-type models (Tatone et al., 2024). Similar mechanotactic effects arise in models where substrate-dependent friction or internal contractility biases cell speeds and drifts (Perthame et al., 9 Jan 2026, Ben-Ami et al., 2023).

4. Macroscopic PDEs: Structure, Limits, and Physical Interpretation

The macroscopic limit of kinetic models generally yields equations of the type: $$\frac{\partial c}{\partial t} = \nabla\cdot\left( M(c) \nabla\left[ k_g \Delta c - f'(c) - \mu_{\mathrm{active}}(c) + \nabla\cdot\ell \right] \right)$$ (Tatone et al., 2024), or (in the classic chemotaxis-diffusion limit): $$\partial_t \rho = \nabla\cdot\big[ D(x) \nabla \rho - \chi(x, S, q)\, \rho \nabla S \big]$$ (Zhigun et al., 2020, Loy et al., 2020), where $D$ reflects anisotropic, environment-dependent motility and $\chi$ is the tactic sensitivity.

Special cases:

• Cahn–Hilliard-type equations: Encode phase separation, interface formation, and “active diffusion” (Tatone et al., 2024).
• Fractional-order models: Long-range Lévy walks and resting times yield anomalous diffusion (space–time fractional PDEs) relevant to immune cell search (Estrada-Rodriguez et al., 2018, Estrada-Rodriguez et al., 2020).
• Contact Guidance/Inhomogeneous Structures: Drift and diffusion tensors depend on local fiber orientation distributions ($q(x, \omega)$), producing complex anisotropic spread, which varies from tissue alignment to random isotropy (Zhigun et al., 2020, Conte et al., 2022, Loy et al., 2019).
• Adhesion-mediated transport: Nonlocal integral fluxes drive aggregation and patterning, modulated by subcellular adhesive state distributions (Zhigun et al., 2023).
• Mechanotaxis: Macroscopic models derived from kinetic equations including frictional substrate interactions or contractile forces explain pattern selection and directional migration in response to mechanical gradients (Perthame et al., 9 Jan 2026, Winkler et al., 17 Feb 2025).

5. Phenotype Structure and Multiscale Approaches

To account for cell-to-cell variability, recent models introduce continuous phenotype variables (e.g., protrusion length, cytoskeletal state) as additional arguments of the cell density $f(t, x, v, y)$, with phenotype-structured kinetic equations incorporating phenotype-dependent sensing and motility rates (Lorenzi et al., 2024). This underpins population heterogeneity and feedback between single-cell and collective behavior, and allows for direct calibration to experimental data, as in stripe migration assays (Lorenzi et al., 2024).

Multiscale closure: Fast-turning/fast-phenotype-change scaling leads to macroscopic limits with drift and diffusion tensors depending on integrals over phenotype distributions and environmental structure.

6. Applications, Computational Aspects, and Boundary Conditions

Typical applications include tissue morphogenesis, wound healing, cancer invasion, immune surveillance, and experimental migration assays (e.g., stripe, organoid, or spheroid migration).

• Boundary conditions: No-flux for cell concentration, reflecting boundaries for velocity distribution, or zero-microstress for concentration gradients ensure mass conservation and physical realism (Tatone et al., 2024, Loy et al., 2019).
• Numerical approaches: Operator-splitting, upwind advection schemes, Monte Carlo sampling, and finite-volume solvers permit robust simulation of kinetic and macroscopic equations, with the multiscale formalism supporting cross-validation between particle-based, kinetic, and continuum models (Lorenzi et al., 2024, Loy et al., 2019, Conte et al., 2022).

7. Outlook and Interconnected Developments

The kinetic equation paradigm has advanced the mechanistic understanding of cell migration by integrating energy-based, stochastic, and multiscale process descriptions. Connections to phase-field models, active matter physics, and biophysical models of actomyosin contractility and focal adhesion further enrich the theoretical framework. Current challenges include rigorous analysis of pattern formation, stability in active gels and mechanotactic systems, and quantitative connection to time-resolved imaging and single-cell tracking data. Open mathematical issues encompass existence and uniqueness for non-linear/nonlocal kinetic PDEs, fractional-diffusion limits, and the impact of complex microenvironmental topology (Tatone et al., 2024, Perthame et al., 9 Jan 2026, Zhigun et al., 2023, Estrada-Rodriguez et al., 2018).

Primary references:

• "Driving forces in cell migration and pattern formation in a soft tissue" (Tatone et al., 2024)
• "A novel derivation of rigorous macroscopic limits from a micro-meso description of signal-triggered cell migration in fibrous environments" (Zhigun et al., 2020)
• "Multiscale analysis of a kinetic equation for mechanotaxis" (Perthame et al., 9 Jan 2026)
• "Phenotype-structuring of non-local kinetic models of cell migration driven by environmental sensing" (Lorenzi et al., 2024)
• "Active gel theory for cell migration with two myosin isoforms" (Winkler et al., 17 Feb 2025)
• "Modelling physical limits of migration by a kinetic model with non-local sensing" (Loy et al., 2019)
• "Space-time fractional diffusion in cell movement models with delay" (Estrada-Rodriguez et al., 2018)
• "A stochastic model for protrusion activity" (Etchegaray et al., 2018)
• "Analysis of a non-local and non-linear Fokker-Planck model for cell crawling migration" (Etchegaray et al., 2017)
• "A non-local kinetic model for cell migration: a study of the interplay between contact guidance and steric hindrance" (Conte et al., 2022)
• "Using a probabilistic approach to derive a two-phase model of flow-induced cell migration" (Ben-Ami et al., 2023)
• "A stochastic modeling framework for single cell migration: coupling contractility and focal adhesions" (Uatay, 2018)
• "Modelling non-local cell-cell adhesion: a multiscale approach" (Zhigun et al., 2023)
• "Multi-cue kinetic model with non-local sensing for cell migration on a fibers network with chemotaxis" (Loy et al., 2020)
• "Macroscopic limit of a kinetic model describing the switch in T cell migration modes via binary interactions" (Estrada-Rodriguez et al., 2020)
• "Kinetic models with non-local sensing determining cell polarization and speed according to independent cues" (Loy et al., 2019)
• "Velocities of Mesenchymal Cells May be Ill-Defined" (Giardini et al., 2023)