Introduction

A common motif in biomaterials is that their elongated constituent elements often align over length-scales much larger than themselves1,2,3. This quasi-long-range orientational order has led to the development of the theory of active nematic liquid crystals4. The study of active nematics — spontaneously flowing systems that exhibit apolar orientational order — has made great strides5,6,7,8 and found wide-ranging applicability, especially in biological settings9,10,11,12. The development of primarily 2-dimensional idealized biophysical experimental systems13,14 and subsequent continuum models5,15 has led to a general understanding of the bulk behavior of these systems4,16. Consequently, the focus has expanded to studying active nematics confined within complicated boundaries17,18,19,20, and the development of strategies to control active flows14,21,22.

Sustained progress into both of these avenues relies on the development of numerical tools capable of flexibly incorporating nontrivial boundaries and inclusions. One such method is Active-Nematic Multi-Particle Collision Dynamics (AN-MPCD)23, which builds on nematic MPCD24 and can reproduce active turbulence in bulk (Fig. 1). Active-Nematic MPCD has proven useful in simulating active turbulence in 3D25. Unlike other commonly used active-nematic solvers7,26, the particle-based nature of MPCD allows for relatively straightforward incorporation of surfaces representing boundaries or inclusions—a feature that has been exploited to study the dynamics of active nematics in porous media27.

Fig. 1: Alternative formulations of cellular activity do not affect bulk active nematic turbulence.
figure 1

Snapshots of local director field nC colored by the scalar order parameter SC for the four different formulations of cellular activity, with an activity parameter of α = 0.15 in a square periodic domain of size ℓsys = 50. “Part.” corresponds to particle-carried activity \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{P}}}}}}}}\) (Eq. (6)); “Cell” to cell-carried activity \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{C}}}}}}}}\) (Eq. (8)); “Mod. Part.” to modulated particle-carried activity \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MP}}}}}}}}\) (Eq. (12)); “Mod. Cell” to modulated cell-carried activity \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MC}}}}}}}}\) (Eq. (11)). Corresponds to frame 75 of Supplementary Movie 1.

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While the particle-based nature of MPCD is advantageous for handling complex geometries, it comes with its own set of challenges: As with any particle-based simulation technique28,29,30,31,32, passive MPCD has local density fluctuations that can affect the dynamics of inclusions33. In the passive case, these are relatively small; however, in the active case these are exacerbated, as exemplified from active Brownian particle simulations and the associated motility-induced phase separation34. The active component of AN-MPCD introduces large number fluctuations that effectively limit the regime over which the algorithm reproduces continuous fields23. In the originally proposed AN-MPCD algorithm, each individual MPCD particle is considered to be an active unit. While sensible from a particle-based perspective, from the continuum perspective this means that the strength of the local active dipole force is proportional to the local fluid density. This implementation ties the local degree of activity to the local density, resulting in a positive feedback: activity leads to regions of high density, which in turn further increases local activity. Studies have considered how activity control affects collective dynamics through the use of light activated active nematics14,22,35 and spatially varying fuel concentration21,36,37.

With the aim of increasing the operational regime of AN-MPCD and improving its suitability for simulating active suspensions, we build upon this by exploring the interplay of local density and the strength of activity through three alternative variant formulations of AN-MPCD. These formulations mitigate density-activity feedback by either: (i) completely untying the strength of activity from the local density, making it uniform for the entire system, (ii) modulating the dependence of activity on density, or (iii) a combination of the previous two methods. Each of the proposed methods quantitatively reproduces the same phenomenology of active nematics (Fig. 1, Supplementary Movie 1). However, the three alternataive formulations greatly reduce the degree of density fluctuations. In particular, the modulated activities decrease and cap the degree of density-induced advection of inclusions to a level only slightly greater than passive MPCD. Furthermore, the effective turbulence regimes are shown to increase by roughly four times with respect to the original particle-based activity implementation. While this work primarily focuses on 2-dimensional extensile active turbulence, matching current experimental work, we verify that the behavior of density fluctuations is qualitatively similar for 3-dimensional and contractile active turbulence. As such, these three activity formulations address the primary short-coming of AN-MPCD for simulating continuous active nematic fields. They not only expand the operational regime of the method, but also improve its suitability for modeling solutes and inclusions, transforming it into a flexible numerical tool for modeling composite active systems. This could include passive colloidal particles or polymers suspended in spontaneously flowing biofluids.

Methods

Passive multi-particle collision dynamics

Multi-Particle Collision Dynamics (MPCD) discretizes a continuous fluid into point-like particles with no steric repulsion, which propagate through streaming and collision steps. Each MPCD particle i of mass mi, position ri, and velocity vi follows a simple Eulerian ballistic streaming step for a time δt to a new position

$${{{{{{{\boldsymbol{r}}}}}}}}_{i}(t+\delta t)={{{{{{{\boldsymbol{r}}}}}}}}_{i}(t)+{{{{{{{\boldsymbol{v}}}}}}}}_{i}(t)\delta t$$
(1)

before undergoing a collision event. Collisions between particles are encoded by the collision operation Ξi,C, which stochastically exchanges momenta and other particle properties within a given MPCD cell C of size a on a square lattice, while ensuring that all appropriate conservation laws are satisfied locally within each cell. The collision operation is applied to each particle within a cell as

$${{{{{{{\boldsymbol{v}}}}}}}}_{i}(t+\delta t)={\left\langle {{{{{{\boldsymbol{v}}}}}}}\right\rangle }_{C}(t)+{{{{{{{\boldsymbol{\Xi }}}}}}}}_{i,C}(t),$$
(2)

where the cellular average \({\left\langle {{{{{{\boldsymbol{v}}}}}}}\right\rangle }_{C}(t)={\sum }_{j}^{{\rho }_{C}}{m}_{i}{{{{{{{\boldsymbol{v}}}}}}}}_{i}(t)/{\sum }_{j}^{{\rho }_{C}}{m}_{i}\) is over the ρC(t) particles currently in cell C. A widely used passive MPCD collision operation for thermalized isotropic fluids is the Andersen collision operator38,39

$${{{{\boldsymbol{\Xi}}}}}_{i,C}^{0} = {{{{\boldsymbol{\xi}}}}}_{i}-{\left\langle {{{{\boldsymbol{\xi}}}}}\right\rangle }_{C}+\left({\underline{\underline{I}}}_{C}^{-1} \cdot \delta {{{{\boldsymbol{{{{\mathcal{L}}}}}}}}}_{{{{\rm{vel}}}}}\right)\times {{{{\boldsymbol{r}}}}}_{i}^{{\prime} },$$
(3)

in which the first two terms represent a stochastic exchange of translational momentum with ξi being a random velocity drawn from the Maxwell-Boltzmann distribution for mass mi and thermal energy kBT and \({\left\langle {{{{{{\boldsymbol{\xi }}}}}}}\right\rangle }_{C}\) being the cellular average of the random velocities. The third term imposes conservation of momentum. The moment of inertia \({\underline{\underline{I}}}_{C}\) is computed from the position of point particles in cell C relative to the cell center of mass \({\left\langle {{{{{{\boldsymbol{r}}}}}}}\right\rangle }_{C}\), where \({{{{{{{\boldsymbol{r}}}}}}}}_{i}^{{\prime} }={{{{{{{\boldsymbol{r}}}}}}}}_{i}-{\left\langle {{{{{{\boldsymbol{r}}}}}}}\right\rangle }_{C}\). The change in angular momentum due to the collision is \({\delta} {{{{\boldsymbol{{{{\mathcal{L}}}}}}}}}_{{{{{{{\rm{vel}}}}}}}}={\sum }_{j}^{{\rho }_{C}}{m}_{j}[{{{{{{{\boldsymbol{r}}}}}}}}_{j}^{{\prime} }\times ({{{{{{{\boldsymbol{v}}}}}}}}_{j}-{{{{{{{\boldsymbol{\xi }}}}}}}}_{j})]\).

The Andersen collision operation (Eq. (3)) highlights the key features of an MPCD operator with each term, namely: It reproduces hydrodynamics by satisfying momentum conservation laws by removing residuals, and it introduces viscous dissipation through stochastic collisions. The collision operation Ξi,C governs the fluid’s material properties and transport coefficients38. Additional material properties can be simulated through more complicated collision operations40,41,42. The most relevant example for this study is the nematic collision operation24. By prescribing an orientation ui to each particle, and a cell-average director nC (calculated via the tensor order parameter), it is possible to devise a nematic collision operations \({{{{{{{\boldsymbol{\Xi }}}}}}}}_{i,C}^{{{{{{{\rm{N}}}}}}}}\) and \({\underline{\underline{R}}}_{i,C}\) to reproduce stochastic linearized nematohydrodynamics.

During each cellular collision, all particle’s orientations are altered by

$${{{{{{{\boldsymbol{u}}}}}}}}_{i}(t+\delta t)={\underline{\underline{R}}}_{i,C}\cdot {{{{{{{\boldsymbol{u}}}}}}}}_{i}(t),$$
(4)

in which \({\underline{\underline{R}}}_{i,C}\) is a rotation matrix representing a nematic orientation collision operation composed of multiple contributions. Inspired by the Andersen collision operation (Eq. (3)), \({\underline{\underline{R}}}_{i,C}\) possesses a component that re-orients particle i by a random orientation drawn from a locally equilibrated Maier-Saupe distribution \(\sim {e}^{U{S}_{C}{({{{{{{{\boldsymbol{u}}}}}}}}_{i} \cdot {{{{{{{\boldsymbol{n}}}}}}}}_{C})}^{2}}\) about the cellular director nC with scalar order SC for an interaction potential U24. Additionally, the rotation of the particles is coupled to their flow because the strain rate and vorticity rotate the orientation of the particles, modeled as slender rods, through the Jeffery’s equation with tumbling parameter λ and hydrodynamic susceptibility χ. The stochastic operation and Jeffery’s equation generate angular momentum \(\delta {{{{{{{\boldsymbol{{{{{{{\mathcal{L}}}}}}}}}}}}}}}_{{{{{{{\rm{N}}}}}}}}={\gamma }_{{{{{{{\rm{R}}}}}}}}\mathop{\sum }_{j}^{{\rho }_{C}}{{{{{{{\boldsymbol{u}}}}}}}}_{j}\times {\dot{{{{{{{\boldsymbol{u}}}}}}}}}_{j}\) for a rotational friction coefficient γR. This is balanced by adding \({{{{{{{\boldsymbol{\Xi }}}}}}}}_{i,C}^{{{{{{{\rm{N}}}}}}}}=-({\underline{\underline{I}}}_{C}^{-1}\cdot \delta {{{{{{{\boldsymbol{{{{{{{\mathcal{L}}}}}}}}}}}}}}}_{{{{{{{\rm{N}}}}}}}})\times {{{{{{{\boldsymbol{r}}}}}}}}_{i}^{{\prime} }\) to the translational collision operation (Eq. (3)). Conservation of angular momentum induces nematic backflow. Passive nematic MPCD (N-MPCD) simulates the Qiang-Sheng model for nematohydrodynamics43.

Activity is introduced through a local cell-based active force dipole, which is applied by adding an active contribution to the collision operation

$${{{{{{{\boldsymbol{\Xi }}}}}}}}_{i}^{{{{{{{\rm{AN}}}}}}}}={\alpha }_{C}\delta t\left(\frac{{\kappa }_{i}}{{m}_{i}}-\frac{{\left\langle \kappa \right\rangle }_{C}}{{\left\langle m\right\rangle }_{C}}\right){{{{{{{\boldsymbol{n}}}}}}}}_{C},$$
(5)

in which αC denotes the local cell dipole strength23. To determine the direction of the active force acting on particle i, cell C is dissected by a plane through the center of mass with normal nC, and \({\kappa }_{i}({{{{{{{\boldsymbol{r}}}}}}}}_{i}^{{\prime} },{{{{{{{\boldsymbol{n}}}}}}}}_{c})=\pm 1\) depending on whether \({{{{{{{\boldsymbol{r}}}}}}}}_{i}^{{\prime} }\) is above or below the plane23. The two terms in Eq. (5) represent the activity that provides individual impulses per unit mass for each particle and a term that removes any residual linear momentum change from the active force term. The induced local force dipole reproduces the bending instability44, unbinding of topological defects45, and active turbulence23. Since the collision algorithm injects energy but conserves momentum, the algorithm simulates wet active nematics23,46.

In each of the four algorithms considered in this article, the activity is included through Eq. (5). The differences enter through the form chosen for the local cell dipole strength \({\alpha }_{C}\left({\rho }_{C}\right)\). The local strength αC(ρC) can be chosen to be linearly proportional to ρC (particle-carried activity23), constant with respect to ρC (cell-carried activity), or modulated by a density kernel (modulated cell-carried and modulated particle-carried activities).

Simulation units and parameters

Results are reported in MPCD units. The unit of length is the MPCD cell size a ≡ 1, the unit of energy is thermal energy kBT ≡ 1, and the unit of mass is particles’ mass m. Since only a single species of fluid particles is considered here, mi = m ≡ 1. As a result, the unit of time is \(a\sqrt{m/{k}_{{{{{{{\rm{B}}}}}}}}T}=1\). The time step size is set to δt = 0.1. The average number of particles per cell is set to \({N}_{{{{{{{\rm{av}}}}}}}}=\left\langle {\rho }_{C}\right\rangle =20\) throughout, where \(\left\langle \cdot \right\rangle\) is the average over the entire system. Due to the point-like nature of MPCD particles with no interaction potentials, the mean particle number per cell Nav is not connected to a volume fraction. Instead, the primary role of the mean particle number per cell is to control fluid transport coefficients38. Due to the selection of a ≡ 1, and m ≡ 1, the number density of a cell ρC is equivalent to the mass density.

The mean field potential is set to U = 10, which puts the system deep in the nematic phase24. The rotational friction coefficient is γR = 0.01, hydrodynamic susceptibility is χ = 0.5 and tumbling parameter is λ = 2 for a flow-aligning nematic. While N-MPCD is composed of point-particles, the mean field parameter U, rotational drag coefficient γR, and tumbling parameter λ are related to an effective aspect ratio47,48. Simulations in 2D are run within a periodic box of size ℓsys × ℓsys = 200 × 200, unless stated otherwise. Simulations in 3D have a volume \({\ell }_{{{{{{{\rm{sys}}}}}}}}^{3}=4{0}^{3}\). Particles are initialized with random positions, thermally sampled velocities and uniform orientation aligned along the \(\hat{y}\) axis. All simulations include 104 warmup steps and run for an additional 5 × 104 data collection steps. A summary of all simulation parameters is provided in Table S1.

Active-Nematic MPCD

In order to employ AN-MPCD, one must choose a form for the activity. In the following section, four formulations are described.

Particle-carried activity

The original AN-MPCD algorithm models activity as though each MPCD particle carries activity23. Therefore, each cell’s activity is set to

$${\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{P}}}}}}}}=\mathop{\sum}_{i=1}^{{\rho }_{C}}{\alpha }_{i},$$
(6)

in which αi is the constant activity prescribed to each particle i. In this definition, the cellular activity is simply the sum of the activities of the particles. Each particle contributes linearly to the cellular activity and so this definition is labeled particle-carried activity. For a system in which every particle carries the same activity (αi = α ∀ i), the cellular activity is simply

$${\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{P}}}}}}}}={\rho }_{C}\alpha ,$$
(7)

which is directly proportional to the number of particles in the cell (Fig. 2 (Part.); dark blue). This is equivalent to assuming that active agents themselves carry the fuel and machinery required for activity. An example system which corresponds to this situation are suspensions of swimming bacteria, for which continuum models make the local activity linearly dependent on bacterial density49 Variations in local density and, consequently, the strength of the local force dipole, depend on the compressibility of the system33.

Fig. 2: Formulations of cellular activity αC as a function of local density ρC.
figure 2

A schematic of the alternative formulations for cellular activity, normalized by the particle activity α. “Part.” corresponds to particle-carried activity \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{P}}}}}}}}\) (Eq. (6)); “Cell” to cell-carried activity \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{C}}}}}}}}\) (Eq. (8)); “Mod. Cell” to modulated cell-carried activity \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MC}}}}}}}}\) (Eq. (11)); “Mod. Part.” to modulated particle-carried activity \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MP}}}}}}}}\) (Eq. (12)). Parameters used in this schematic are \(\left\langle {\rho }_{C}\right\rangle =5\), σp = 1, and σw = 1. At ρC = 0, cell- and modulated cell-based activity take a value of 0, indicated by the marker.

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The \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{P}}}}}}}}\) activity formulation (Eq. (7)) is effective in simulating active nematic liquid crystals23, satisfying the theoretically expected scalings of active nematics for a range of parameters45,50. However, unlike continuum solvers of the active nematic equations of motion7,51,52 which typically assume incompressibility, this particle-based approach possesses a large degree of density fluctuations23. In some cases, such as in modeling bacteria suspension that do exhibit large variations in swimmer density10,53, the fluctuations may be desired, while in others they may hamper studies of systems in which density-induced pressure gradients may affect the physics, such as suspensions of colloid54 or polymer solutes55,56. While it was shown that density fluctuations can be mitigated by limiting the activity regime (i.e., providing an upper bound for α)23, this might not always be desirable when reproducing experimental data. The fluctuations result from a positive feedback, in which high density regions result in large local activities, which then create more higher density regions.

Cell-carried activity

To break the positive feedback, activity should not increase linearly with density. In this section, the activity is made nearly constant. The formulation considered here normalizes \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{P}}}}}}}}\) by ρC to remove the direct density dependence from Eq. (6) via the redefinition of each cell’s activity as

$${\alpha }_{C}^{C}=\left\{\begin{array}{ll}{\left\langle {\alpha }_{i}\right\rangle }_{C}\quad &{{{{{{\rm{if}}}}}}}\,{\rho }_{C} \, > \, 0\\ 0\quad \hfill& \;\,{{{{{{\rm{if}}}}}}}\,{\rho }_{C}=0\end{array}\right.,$$
(8)

where \({\left\langle \alpha \right\rangle }_{C}=\left({\sum }_{i}^{{\rho }_{C}}{\alpha }_{i}\right)/{\rho }_{C}={\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{P}}}}}}}}/{\rho }_{C}\). For a single population of particles with constant particle activity (αi = α ∀ i), cell activities are constant and

$${\alpha }_{C}^{C}=\alpha$$
(9)

everywhere, except in the case of empty cells since there can be no force dipole in an empty cell. In contrast to \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{P}}}}}}}}\), this homogenizes the cellular activity throughout the simulation: every non-empty MPCD cell has the same activity, regardless of the local density.

In this version, the activity is carried by cells rather than by the particles, and so this choice is referred to as cell-carried activity (Fig. 2 (Cell.); red). This is equivalent to assuming the fuel/microscopic machinery for activity is equally dispersed throughout the system. In experimental microtubule/kinesin bundle systems, measures of activity with respect to ATP concentration shows a sigmoidal form: At small concentrations, activity rises linearly with ATP, but plateaus to a constant value at high concentrations57,58. Thus, particle-carried activity might be appropriate to replicate experiments with low concentrations of ATP, while cell-carried activity might become a better model as ATP concentration is increased.

Modulated activities

While cell-carried activity \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{C}}}}}}}}\) breaks the positive feedback, it does not proactively discourage high densities. To more closely match the common continuum-limit assumption of constant density16 by further mitigating the activity-induced density variations, activity could be formulated to impose a negative feedback at high densities. To do this, a modulation function \({{{{{{{\mathcal{S}}}}}}}}_{C}\) is introduced as a multiplicative factor on \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{P}}}}}}}}\) or \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{C}}}}}}}}\). This modulation function depends on the local density ρC and is chosen to decrease to zero as ρC → ∞ but only activating at high densities, leaving the small density behavior intact. Thus, a sigmoidal modulation function

$${{{{{{{\mathcal{S}}}}}}}}_{C}=\frac{1}{2}\left(1-\tanh \left(\frac{{\rho }_{C}-\left\langle {\rho }_{C}\right\rangle \left(1+{\sigma }_{{{{{{{\rm{p}}}}}}}}\right)}{\left\langle {\rho }_{C}\right\rangle {\sigma }_{{{{{{{\rm{w}}}}}}}}}\right)\right)$$
(10)

is selected. The modulation function \({{{{{{{\mathcal{S}}}}}}}}_{C}\left({\rho }_{C};{\sigma }_{{{{{{{\rm{p}}}}}}}},{\sigma }_{{{{{{{\rm{w}}}}}}}}\right)\) compares a cell’s instantaneous density ρC(t) to the system-wide mean density \(\left\langle {\rho }_{C}\right\rangle\) and returns a value in the interval (0, 1). The two parameters σp and σw set the position and width of the sigmoidal drop. In particular, σp sets the position of the sigmoid midpoint, with a value of σp = 0 placing the midpoint at \(\left\langle {\rho }_{C}\right\rangle\), while σp > 0 (σp < 0) shifts the midpoint to higher (lower) densities.

The modulation can be applied to either particle- or cell-carried activity. In the case of the modulated cell-carried activity,

$${\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MC}}}}}}}}={\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{C}}}}}}}}{{{{{{{\mathcal{S}}}}}}}}_{C}\left({\rho }_{C};{\sigma }_{{{{{{{\rm{p}}}}}}}},{\sigma }_{{{{{{{\rm{w}}}}}}}}\right)$$
(11)

Because \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{C}}}}}}}}\) is independent of density, the modulation acts like a switch lowering the activity at densities above \({\sigma }_{{{{{{{\rm{p}}}}}}}}\left\langle {\rho }_{C}\right\rangle\) (Fig. 2 (Mod. Cell); yellow).

Similarly, the modulation can be applied to create a modulated particle-carried activity

$${\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MP}}}}}}}}={\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{P}}}}}}}}{{{{{{{\mathcal{S}}}}}}}}_{C}\left({\rho }_{C};{\sigma }_{{{{{{{\rm{p}}}}}}}},{\sigma }_{{{{{{{\rm{w}}}}}}}}\right).$$
(12)

The activity formulation \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MP}}}}}}}}\) appears qualitatively similar to a Maxwell-Boltzmann distribution, mimicking the linear behavior of \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{P}}}}}}}}\) for small ρC, but decaying to zero at large ρC (Fig. 2 (Mod. Part.); cyan). One way of interpreting the modulation is that the efficacy of the activity to produce active stresses decreases as the density passes some value, eventually falling to 0 for high densities. In the microtubule/kinesin system, this could be expected if the concentration of motors becomes too high59.

Activity parameters

In this study, we limit our consideration to a single species of active fluid particles by setting αi = α ∀ i and considing extensile activity (α > 0), unless stated otherwise. The sigmoidal modulation parameters are set to σp = 0.4 and σw = 0.5, which are found to give optimal density behavior without lowering effective activity (see Fig. S1).

Results

Previous work has verified that particle-carried activity (\({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{P}}}}}}}}\)) reproduces active turbulence23. It was demonstrated that the particle-carried activity algorithm possesses four distinct regimes as a function of activity: for small activities, α ≲ αeq ≈ 10−3, the thermostat dominates over activity. In the second regime, αeq ≲ α ≲ αturb ≈ 0.02, there is enough active stress to form bend walls within finite sized systems. In the third regime, αturb ≲ α ≲ α†, the active stress is sufficient for pair creation events to occur, driving the system into active turbulence. Finally, for α ≳ α† density fluctuations become large. The following sections will demonstrate that each of these regimes is still present in the modulated methods.

Qualitatively, all four activity formulations reproduce active turbulence (Fig. 1; Supplementary Movie 1). In each, the transient dynamics of active turbulence onset is the same (Supplementary Movie 2): The activity-induced hydrodynamic instability60 generates narrow kink walls61,62 that are perpendicular to the initially uniform director. Along the kink walls, active forces drive viscometric flows52. At sufficiently high activity, the free energy cost of the bend walls is large enough that topological defect pairs unbind, with defect core sizes comparable to the MPCD cell size a. The + 1/2 defects are self-motile, moving rapidly through the system, while the − 1/2 defects are only passively advected by the flows13. This description holds for all four activity formulations, with the distance separating defects and the speed of the + 1/2 qualitatively similar in each (Fig. 1).

Scaling of active turbulence

A common feature of active nematic turbulence is the presence of unbound ± 1/2 nematic defects63. In 2D, the defects with the lowest energetic cost compatible with nematic symmetry are the ± 1/2 defects, corresponding to windings of ± π/264. The fact that active turbulence coincides with the unbinding of topological defects is not a coincidence, as defect dynamics drive active turbulence5. Particle-carried AN-MPCD successfully exhibits defect pair creation and annihilation events, producing an active length scale (Fig. 3). The active length scale is found by computing the root-mean square mean defect separation \({\ell }_{d}={\rho }_{d}^{-1/2}\) from the mean number density of defects ρd. Likewise, the cell-carried activity and both modulated formulations successfully reproduce defects and the expected scaling behavior.

Fig. 3: Defect separation.
figure 3

Steady-state defect density ρd is used to compute the root-mean square defect separation \({\ell }_{d}={\rho }_{d}^{-1/2}\) for all four formulations of cellular activity. The dashed line indicates a scaling of ℓd ~ α−1/2, which is the theoretically expected scaling for active nematic turbulence. ("Part.'', Eq. (6), blue; “Cell'', Eq. (8), red; “Mod. Part.'', Eq. (12), cyan; “Mod. Cell'', Eq. (11), yellow).

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For all activity formulations, the defect density is zero until a critical activity (αturb) above which turbulence occurs. The αturb value varies slightly between formulations (Fig. 3), with the onset for the modulated methods occurring only at a slightly higher activity. Once the critical activity is reached, the defect density increases with activity, and the defect separation length decreases as ℓd ~ α−1/2 (Fig. 3; dashed line). Measured exponents for each formulation (Table S2) demonstrate that the modulated methods reproduce the theoretical expectations for the nematic length scale in active turbulence45,50. This demonstrates the scaling of active turbulence is not affected by the choice of cell activity formulation.

Likewise, the spontaneous active flows scale as vav ~ α1/2 65. The speed resulting from active turbulence for each formulation (Fig. 4a) is obtained by measuring the average root-mean square flow speed vav and subtracting the residual thermal speed \({v}_{{{{{{{\rm{av}}}}}}}}^{0}=\mathop{\lim }_{\alpha \to 0}{v}_{{{{{{{\rm{av}}}}}}}}\ne 0\). This yields two distinct scaling regimes: At lower activities, α ≲ αturb, system-spanning bend walls drive slow but coherent flows in the system. This results in a scaling that is consistent with vav ~ α (Table S2). At higher activities when the system enters into active turbulence (α ≳ αturb), all formulations match the theoretical scaling, as indicated by the dashed line in Fig. 4a and Table S2, with the exception of particle-carried activity at very large activities. Both regimes exist for all four activity formulations. All activity formulations reproduce the expected flow speed scaling laws in the active turbulence regime and within their operational regime.

Fig. 4: Velocity scalings of active turbulence.
figure 4

a The velocity contribution due to active flows, \({v}_{{{{{{{\rm{av}}}}}}}}-{v}_{{{{{{{\rm{av}}}}}}}}^{0}\), for all four formulations of cellular activity. The dashed line indicates the theoretically expected scaling of vav ~ α1/2. b The velocity length scale of active turbulence ℓv in the turbulence regime. The dotted line indicates the theoretically expected scaling of ℓv ~ α−1/2. ("Part.'', Eq. (6), blue; “Cell'', Eq. (8), red; “Mod. Part.'', Eq. (12), cyan; “Mod. Cell'', Eq. (11), yellow).

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For contractile activities, cell-carried and modulated activities flow speeds show agreement with theoretical predictions in the active turbulence regime \({\alpha }_{{{{{{{\rm{turb}}}}}}}}\lesssim \left\vert \alpha \right\vert \lesssim {\alpha }_{{{{\dagger}}} }\) (Fig. S2a). However, particle-carried activity only agrees with theory up to intermediate activities, and at higher activities becomes non-monotonic. This is due to a breakdown in the algorithm for contractile activity at large magnitudes of cellular activity, \(\left\vert {\alpha }_{{{{{{{\rm{C}}}}}}}}\right\vert\): In contractile AN-MPCD, particles receive impulses towards the center of mass plane, and large cellular activities result in these particles crossing over the plane defined in Eq. (5) in a single timestep. Such cells are effectively extensile, but with a lower effective magnitude of cellular activity since part of the cellular activity is used to cross the plane. This is only observed to occur for particle-carried activity.

The radial velocity auto-correlation functions, \({C}_{v,v}(R)={\left\langle {{{{{{\boldsymbol{v}}}}}}}({{{{{{\boldsymbol{r}}}}}}})\cdot {{{{{{\boldsymbol{v}}}}}}}({{{{{{\boldsymbol{r}}}}}}}+R{{{\hat{{{{\boldsymbol{r}}}}}}}})\right\rangle }_{{{{{{{\boldsymbol{r}}}}}}},t}/{\left\langle {v}^{2}({{{{{{\boldsymbol{r}}}}}}})\right\rangle }_{{{{{{{\boldsymbol{r}}}}}}},t}\) (Fig. S3) are fit by an exponential decay to measure the velocity length scale of active turbulence, ℓv (Fig. 4b). For αeq ≲ α ≲ αturb, the active flow driven by the nascent nematic deformations of the bend walls are largely cohesive, leading to high ℓv. Furthermore, these flows are parallel to the bend walls, which increases their amplitude, but crucially, not their wavelength. The characteristic decorrelation length scale is largely independent of the particle activity α in this regime. On the other hand in the turbulent regime, the length scale decays as ℓd ~ α−1/2 (Table S2), which matches the theoretical prediction45,50.

While flow speeds and both director and velocity length scales are an important component of active nematic flows, it is also important to consider the enstrophy spectra. Unlike inertial turbulence, it has been shown that active nematics do not possess scale invariance66. The active nematic length scale ℓα is representative of the characteristic vortex size within active turbulence, and corresponds to a wavenumber kα that maximizes enstrophy spectra (Fig. S4 for α = 0.1). At small wavenumbers, with k < kα, theoretical expectations show that enstrophy rises linearly with ~ k+1. Simulations within the turbulence regime αturb ≲ α ≲ α† are consistent with this scaling (Table S2); however, low wavenumbers (corresponding to length scales approaching the system size) possess the greatest statistical uncertainty. At high wavenumbers, the theoretically expected scaling of ~ k−2 is recovered for all formulations (Table S2). At the very largest wavenumbers (k ≥ ka ≃ 2π/3a), a secondary peak appears, which is an artifact of vorticity being calculated on the MPCD cell lattice23.

In conclusion, regardless of the chosen activity formalism, AN-MPCD successfully reproduces the theoretically expected scaling laws for active turbulence within its operational regime. However, the onset of turbulence varies depending on formalism, indicating that the activity operational regimes depend on the implementation of activity.

Density variation

As a particle-based model of active matter, AN-MPCD exhibits density fluctuations28,29,30,31. Furthermore, since its equation of state corresponds to an ideal gas33,38, MPCD simulates a compressible fluid67,68. Together these produce density fluctuations, which in turn induce pressure gradients and resulting flows. If significant, these flows may affect the dynamics of suspended solutes, such as colloids, tracers and polymers. This section will show that the modulated activities greatly reduce density fluctuations when compared to particle-carried activity by characterizing the behavior of density fluctuations and the corresponding density-induced gradients, demonstrated in Fig. 5; Supplementary Movie 3.

Fig. 5: Alternative formulations of cellular activity mitigate density fluctuations.
figure 5

Snapshots of the local density field ρC for the four different formulations of cellular activity, with an activity parameter of α = 0.15 in a square periodic domain of size ℓsys = 50. Corresponds to frame 75 of Supplementary Movie 3. ("Part.'', Eq. (6); “Cell'', Eq. (8); “Mod. Part.'', Eq. (12); “Mod. Cell'', Eq. (11)).

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In the low activity limit, the density distributions are Gaussian about \(\left\langle {\rho }_{C}\right\rangle\) (Fig. S5). As the activity is increased, the distributions skew, producing high density tails. While qualitatively all density distributions behave similarly, a quantitative analysis reveals differences. While the mean of the distributions is fixed by the number of particles in the simulation, the median is constantly ρC = 20 at low activity for all activity formulations (Fig. 6a). In each formulation, the cell population median starts to decrease in the turbulent regime, α ≳ αturb (Fig. 6a). While all four formulations exhibit this behavior, particle-carried activity \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{P}}}}}}}}\) exhibits the largest drop. Since particle number is conserved, the smaller cell population median in the active turbulent regime arises because most cells lose particles to a relatively few cells that drastically increase their local density.

Fig. 6: Modulated formulations for cellular activity mitigate density fluctuations.
figure 6

a The median cell population ρC for all four formulations of cellular activity. Modulated methods result in the system median dropping below the mean density within the turbulence regime. Inset: The standard deviation of cell population \({{\sigma}_{{\rho}_{C}}}\). b The fraction of empty cells with ρC = 0. Particle-carried activity in particular has a large fraction of empty cells at the highest activities, while alternative formulations result in a more homogeneous system. c The density correlation length ℓρ, representative of the width of high density nematic bands throughout the system. Density bands become more dense and thinner as activity is raised, but alternative formulations mitigate this effect. The dotted line indicates a scaling of ℓρ ~ α−1. Inset: The far-field value of the density correlation function Cρ,ρ(R → ∞), indicating the degree of density homogeneity. Modulated methods result in a more homogeneous system. d The temporal average of the instantaneous spatial maxima of local density, \({\left\langle \max {\left[{\rho }_{C}({{{{{{\boldsymbol{r}}}}}}},t)\right]}_{{{{{{{\boldsymbol{r}}}}}}}}\right\rangle }_{t}\). ("Part.'', Eq. (6), blue; “Cell'', Eq. (8), red; “Mod. Part.'', Eq. (12), cyan; “Mod. Cell'', Eq. (11), yellow).

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This results in an associated increase in the standard deviations (Fig. 6a inset). The standard deviations are constant at low activities but rise in the active-turbulence regime. This is true in all formulations of activity; however, it is once again most pronounced for the particle-carried activity \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{P}}}}}}}}\) (Fig. 6a inset). The second largest standard deviation is for cell-carried activity \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{C}}}}}}}}\), while the two modulated versions, \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MP}}}}}}}}\) and \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MC}}}}}}}}\), are the smallest and relatively indistinguishable. Thus, the modulation decreases density variation, as intended.

It is useful to consider the extrema of low and high densities. At the lowest of densities, the fraction of completely empty cells (ρC = 0) raises sharply above α ≳ αturb (Fig. 6b). Because observables are computed from cell averages, a large fraction of empty cells marks the break down of the MPCD algorithm. In this regard, particle-carried activity has an order of magnitude larger fraction of empty cells when compared to all other formalisms at α ≳ αturb activity values. In particular, the modulated particle-carried activity \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MP}}}}}}}}\) has the lowest fraction of empty cells—each more than an order of magnitude smaller than particle-carried activity.

In the other extreme, in the turbulent regime high density cells form thin, high-density nematic bands69 (Supplementary Movie 3). The radial density auto-correlation function \({C}_{\rho ,\rho }(R)={\left\langle {\rho }_{C}({{{{{{\boldsymbol{r}}}}}}}){\rho }_{C}({{{{{{\boldsymbol{r}}}}}}}+ R{{{\hat{{{{\boldsymbol{r}}}}}}}})\right\rangle }_{{{{{{{\boldsymbol{r}}}}}}},t}/ {\left\langle {{\rho }_{C}}^{2}({{{{{{\boldsymbol{r}}}}}}})\right\rangle }_{{{{{{{\boldsymbol{r}}}}}}},t}\) characterizes the width of these bands (Fig. S6) through the decorrelation length ℓρ (Fig. 6c). As the system enters its turbulent regime and bands form, increasing activity decreases the density decorrelation lengths (Fig. 6c). The density decorrelation length scales roughly inverse to activity as ℓρ ~ α−1 (see Table S2), indicating nematic bands become increasingly thinner. While this behavior is shared by all activity formulations, particle-carried activity leads to the thinnest bands, whereas the modulated methods provide the widest bands (with roughly twice the decorrelation length when compared to particle-carried activity). Due to MPCD particles having no steric repulsion, these bands are allowed to freely propagate without the formation of clusters as would occur in motility-induced phase separation34.

Complementing the decorrelation length is the far-field value of the correlation function, Cρ,ρ(R → ∞) (Fig. 6c; inset). The far-field decorrelation describes how pronounced the density fluctuations are, providing an estimate of how dense bands are compared to the diluted surroundings. Not only does particle-carried activity have the thinnest bands (Fig. 6c), it also has the largest density difference. On the other hand, the modulated methods \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MP}}}}}}}}\) and \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MC}}}}}}}}\) have the least particle accumulation within bands. This behavior is further verified by considering the time average of the instantaneous maximum cell density, \({\left\langle \max {\left[{\rho }_{C}({{{{{{\boldsymbol{r}}}}}}},t)\right]}_{{{{{{{\boldsymbol{r}}}}}}}}\right\rangle }_{t}\), which measures how dense the peak of the nematic bands are for each activity formulation (Fig. 6d). Consistent with all other metrics of density variation, particle-carried activity \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{P}}}}}}}}\) has significantly larger maximum instantaneous densities compared to the modulated methods. This indicates that, while density bands are present in all activity formulations, the modulated activities lead to attenuated bands that are wider and less concentrated, and thus to more homogeneous systems.

The behavior of density fluctuations in 3-dimensional extensile AN-MPCD active turbulence (Fig. S7) is broadly consistent with 2D results for all four activity formulations. The primary difference is that particle-carried and cell-carried activity in 3D exhibit more similar levels of density fluctuations, in particular in median cell density (Fig. S7a), standard deviations of cell density (Fig. S7a inset), and mean instantaneous maximal cell density (Fig. S7c). However, both modulated activities result in a more homogeneous density fields, which is consistent with the findings in 2D though with less variation at the extremes, including less empty cells (Fig. S7b) and smaller maximal populations (Fig. S7c). Contractile activities (α < 0) in 2-dimensional active turbulence (Fig. S2b–d) show similar behavior to extensile turbulence. There are some particular exceptions however: Particle-carried activity exhibits non-monotonic behavior throughout, due to the breakdown of the algorithm from large cellular activity magnitudes \(\left\vert {\alpha }_{{{{{{{\rm{C}}}}}}}}\right\vert\). For modulated activities, rather the median density decreasing with activity (as occurred in the extensile case; Fig. 6a), they are more constant, ticking up slightly at the highest activities. Furthermore, the fraction of empty cells (Fig. S2c) is generally larger for all formulations compared with extensile, although the mean instantaneous maximal population (Fig. S2d) are smaller for modulated activities while larger for cell-carried activity.

Overall, these results illustrate how cell-carried activity \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{C}}}}}}}}\) reduces density variation compared to the original implementation of particle-carried activity \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{P}}}}}}}}\); however, it is the modulated activities \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MP}}}}}}}}\) and \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MC}}}}}}}}\) that produce the most homogeneous solvents.

The activity-driven variation of density can be further quantified by measuring the giant number density fluctuations10,70,71. The standard deviation of density in local regions of increasing size is measured as a function of the average number of particles σρ(ρ) (Fig. S8). The resulting standard deviations are fit to a power law, \({\sigma }_{\rho }({\rho })=A{\rho }^{\nu }\), which in equilibrium scales as \({\sigma }_{\rho } \sim {\rho }^{1/2}\) according to the central limit theorem. However, as for many non-equilibrium active particle systems10,60,72,73,74, ν → 1 in the turbulent regime irrespective of the activity formulation (Fig. 7a).

Fig. 7: Giant number fluctuations (GNF) analysis reveals algorithm regimes.
figure 7

a The scaling ν of giant number fluctuations, σρ = Aρν. Each of these methods has similar scaling, with a shift for the modulated formulations. b The prefactor A of fits to the giant number fluctuations, σρ = Aρν. In the passive regime α ≲ αeq these remain at a unit value. Once in the nematic bend regime αeq ≲ α ≲ αturb these begin to drop, with a minimum when turbulence forms at α ≃ αturb. In the turbulence regime αturb ≲ α the prefactor rises until it exceeds unit value when α ≳ α†. c The effective turbulence regime αturb ≲ α† for the four formulations of cellular activity, obtained from (b). These represent the activity range for which the algorithm successfully reproduces active turbulence, which begins at αturb when the hydrodynamic bend instability can fit within the system and ends at α† when the fluctuations vastly exceed the equilibrium limit. ("Part.”, Eq. (6), blue; “Cell”, Eq. (8), red; “Mod. Part.”, Eq. (12), cyan; “Mod. Cell”, Eq. (11), yellow).

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While the phrase giant number fluctuations refers to the increasing of ν above 1/2, it does not necessarily refer to the actual magnitude of the variations. Indeed, what matters for solute dynamics is not necessarily the exponent but rather the local amplitude of the variations, which is quantified by the prefactor A (Fig. 7b). This prefactor can be interpreted as A = σρ(1) is therefore a measure of local density fluctuations in the limit of a single particle in a cell. The behavior of A is found to be a particularly clear descriptor of the different regimes of activity for each formulation. At low activities (α < αeq), the source of fluctuations is the thermostat and the prefactor is identical for all formulations of activity (Fig. 7b). At higher activities (αeq ≲ α ≲ αturb), the active force overcomes the thermostat and the system begins to exhibit bend walls on scales comparable to the system size with their associated deterministic flows (Fig. 7b). As flows become more deterministic, individual particles generally travel along the direction of the active forcing, surpressing thermal fluctuations (A < 1). As the activity is further increased (αturb ≲ α ≲ α†), the bend instability is triggered and the active turbulence competes against coherent flows increasing fluctuations, and causing A to rise (Fig. 7b). At even greater activities (α ≳ α†), the amplitude A > 1 indicates substantial non-equilibrium effects to the density (Fig. 7b). At these high activities, the activity-induced density effects are strong, and the algorithm no longer reproduces a continuum fluid on large length scales. This occurs first in particle-carried activity and last in the modulated activities. From this we can infer the critical activities for the bend instability αeq, the onset of active turbulence αturb, and the onset of substantial density fluctuations α†. In particular, αturb and α† serve as bounds for each activity formulation’s operational turbulence regime (Fig. 7c and Table S3). The two modulated activities \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MP}}}}}}}}\) and \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MC}}}}}}}}\) are seen to have the widest operational regime, roughly four times wider than the particle-carried activity \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{P}}}}}}}}\) (Fig. 7c).

Density gradients

While the attenuation of giant-number fluctuations by modulated activities is promising, density bands may still induce local fluxes, which affect the behavior of solutes or suspended particles. As the MPCD equation of state corresponds to an ideal gas, Fick’s first law states that density-induced diffusive fluxes are driven by density gradients, ∇ρ. Hence, the magnitude of density gradients, \(\left\vert {{{{{{\boldsymbol{\nabla }}}}}}}\rho \right\vert\), measures the contribution of local density variations to the flux. The distribution of \(\left\vert {{{{{{\boldsymbol{\nabla }}}}}}}\rho \right\vert\) for a fixed activity α = 0.1 (Fig. 8a) reveals that all activity formulations present Maxwell-Boltzmann-like behavior. The particle-carried activity has the longest tail, while the two modulated activities have the shortest tails. The modulated activity formulations are very close to the passive case, which is a Maxwell-Boltzmann distribution with standard deviation σ = 0.85 ± 0.05.

Fig. 8: Modulated formulations for cellular activity mitigate density-induced drift.
figure 8

a Distributions of the magnitude of density-induced drift, \(\left\vert {{{{{{\boldsymbol{\nabla }}}}}}}\rho \right\vert\), for all four formulations of cellular activity at a fixed activity of α = 0.1 and for passive N-MPCD. b The mean magnitude of density-induced drift, \(\left\langle \left\vert {{{{{{\boldsymbol{\nabla }}}}}}}\rho \right\vert \right\rangle\). Inset: The standard deviation of density-induced drift normalized by the mean. c The magnitude of density-induced drift \(\left\vert {{{{{{\boldsymbol{\nabla }}}}}}}\rho \right\vert\) as a function of local density, \(\left\vert {{{{{{\boldsymbol{\nabla }}}}}}}{{\rho}} \right\vert\,({{\rho}_{C}})\) at a fixed activity of α = 0.1 and for passive N-MPCD. The beginnings of plateaus of maximum density-induced drift are seen for the modulated methods, capping the effect density fluctuations can have on solutes. ("Part.'', Eq. (6), blue; “Cell'', Eq. (8), red; “Mod. Part.'', Eq. (12), cyan; “Mod. Cell'', Eq. (11), yellow).

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Lower activities (α = 0.08) have narrower distributions, with less pronounced differences between the modulated activities and cell-carried activity, although particle-carried activity maintains its significantly wider tail (Fig. S9a). Meanwhile higher activities (α = 0.3) have much wider distributions, although the modulated activities still have the shortest tails (Fig. S9b). As activity increases, the mean and width of the distributions increase (Fig. 8b, b inset), indicating that density-induced drift should be expected to become larger and more widely distributed. Particle-carried activity consistently presents significantly wider tails, and therefore, larger expected fluxes than the other activity choices. The modulated activities generally have the lowest gradients, barely increasing beyond the equilibrium limit of α → 0 for activities α ≲ 0.1 (Fig. 8b).

To further investigate why this is the case, the magnitude of density gradients as a function of local density is measured for a fixed activity (α = 0.1) and compared with the passive case (Fig. 8c). While the passive case remains constant for all densities, both particle- and cell-carried activities exhibit monotonically increasing \(\left\vert {{{{{{\boldsymbol{\nabla }}}}}}}\rho \right\vert\) as a function of local density. However, the modulated activities begin to plateau to a value only slightly beyond the passive limit as \({\rho }_{C}\to 1.5\left\langle {\rho }_{C}\right\rangle\). As with the distributions of \(\left\vert {{{{{{\boldsymbol{\nabla }}}}}}}\rho \right\vert\), this behavior still occurs for lower activities (Fig. S10a), but is less pronounced. Meanwhile for higher activities (Fig. S10b), the modulated activties exhibit a fully formed plateau. This indicates that the modulated activities \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MP}}}}}}}}\) and \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MC}}}}}}}}\) have comparable Fickian fluxes to the passive limit.

The behavior of Fickian fluxes in 3-dimensional active turbulence (Fig. S7d) are qualitatively similar to the 2D case (Fig. 8a). Modulated activities in 3D have smaller means and standard-deviations in the magnitudes of density-gradient driven flows. Likewise, contractile activities in 2-dimensional active turbulence exhibit similar Fickian fluxes to that of extensile turbulence (Fig. S2e), although the mean magnitude of density-induced fluxes is smaller for modulated activities while larger for cell-carried activity.

The density gradients suggest that Fickian fluxes in the modulated activities should not affect solutes substantially more than in passive MPCD. To confirm this, point-like tracer particles are suspended within the active nematic solvent, whose trajectories can be recorded and directly analyzed. Tracer particles participate in the collision operation and have the same mass mT = m as the solvent, but are not active αT = 0. The velocity of tracer particles vT is correlated to the local fluid fields at the level of MPCD cells. In particular, the correlation between tracer velocity and terms proportional to the active forcing f = (∇ ⋅ nC)nC and proportional to the local density gradients \({{\mathbf{{\mathcal{J}}}}}={{{{{{\boldsymbol{\nabla }}}}}}}\rho\) are measured. This is done using the Pearson correlation coefficient (PCC), \({{{{{{\mathcal{P}}}}}}}_{X,Y}=\left\langle \left(X-\left\langle X\right\rangle\right) \left(Y-\left\langle Y\right\rangle \right) \right\rangle /{\sigma }_{X}{\sigma }_{Y}\) between variables X and Y with standard deviations \({\sigma }_{X}={\left\langle X^2-\left\langle X\right\rangle^2 \right\rangle }^{1/2}\). The PCC is used to reveal whether the magnitudes of the local density gradient \(\left\vert {{{{{{\mathcal{J}}}}}}}\right\vert =\left\vert {{{{{{\boldsymbol{\nabla }}}}}}}\rho \right\vert\) or active forcing \(\left\vert f\right\vert =\left \vert {{{{{{\boldsymbol{\nabla }}}}}}}\cdot {{{{{{{\boldsymbol{n}}}}}}}}_{C}\right\vert\) correlate to tracer speed \(\left\vert {v}_{T}\right\vert\). Likewise, angular PCC determines whether tracer particles are more biased in the direction of the density gradients \(\hat{{{{{{{\mathbf{{\mathcal{J}}}}}}}}}}={{{{{{\boldsymbol{\nabla }}}}}}}\rho /\left\vert {{{{{{\mathcal{J}}}}}}}\right\vert\) or active forcing \(\hat{{\boldsymbol{f}}}=({{{{{{\boldsymbol{\nabla }}}}}}}\cdot {{{{{{{\boldsymbol{n}}}}}}}}_{C}){{{{{{{\boldsymbol{n}}}}}}}}_{C}/\left\vert f\right\vert\).

The angular correlation between the density gradients and tracer velocity, \({{{{{{\mathcal{P}}}}}}}_{{\hat{v}}_{T},{\hat{{{{{{{\mathcal{J}}}}}}}}}}\), indicates complete decorrelation, irrespective of activity (Fig. S11a). This indicates that tracers are not biased in the direction of density gradients. On the other hand, the correlation of the magnitude between the tracer particle velocity and the density gradient, \({{{{{{\mathcal{P}}}}}}}_{\left\vert {v}_{T}\right\vert, \left\vert {{{{{{\mathcal{J}}}}}}}\right\vert}\), is decorrelated at low activities but, within the active turbulence regime, it is not (Fig. 9a; α ≳ αturb). Within the turbulence regime, the various activity formulations behave quite differently. The Pearson correlation coefficient of the magnitude for particle-carried activity strongly increases with activity. This positive correlation indicates that tracer particles are more likely to travel faster when the density gradient is large, which is expected due to the formulation of particle-carried activity. In contrast, all other activity choices result in negative Pearson correlation coefficients that decrease with activity, indicating that tracer particles move less quickly in regions with greater density gradients. This is particularly striking for the modulated activity variants; Cells with higher magnitudes of density-gradient induced fluxes generally have higher densities (Fig. 8c). At the same time, the modulation reduces the local activity in high density regions, lowering the speeds of particles in these regions. Together, this results in a negative correlation between the local density gradient and the tracer particle velocities. While the strength of correlation and anti-correlation can reach ± 0.2 at the highest activities, the magnitude of correlation coefficients remains ≈ 0.05 within the operational regimes identified in Fig. 7c. This is comparable to the uncertainty on the correlation coefficients of the passive MPCD case. Taking both the angular and magnitude PCC into account, the modulated activities are indeed counteracting the activity-density feedback: Tracer particles no longer encounter an enhancement to their speed and have no directional bias in regions of high density gradients.

Fig. 9: Pearson correlation coefficients (PCC) of tracer particle velocity to cell quantities have non-trivial dependency on activity formulation.
figure 9

a PCC \({{{{{{\mathcal{P}}}}}}}_{\left\vert {v}_{T}\right\vert, \left\vert {{{{{{\mathcal{J}}}}}}}\right\vert}\) of tracer particle speeds \(\left\vert {v}_{T}\right\vert\) compared to the magnitude of density-induced drift \(\left\vert {{{{{{\mathcal{J}}}}}}}\right\vert =\left\vert {{{{{{\boldsymbol{\nabla }}}}}}}\rho \right\vert\). b PCC \({{{{{{\mathcal{P}}}}}}}_{\left\vert {v}_{T}\right\vert, \left\vert f\right\vert}\) of tracer particle speeds \(\left\vert {v}_{T}\right\vert\) compared to the magnitude of active force \(\left\vert f\right\vert =\left\vert {{{{{{\boldsymbol{\nabla }}}}}}}\cdot {{{{{{{\boldsymbol{n}}}}}}}}_{C}\right\vert\). Inset: PCC \({{{{{{\mathcal{P}}}}}}}_{{\hat{v}}_{T},\hat{f}}\) of tracer particle velocity directions \({\hat{v}}_{T}\) compared to the direction of active force \(\hat{f}=({{{{{{\boldsymbol{\nabla }}}}}}}\cdot {{{{{{{\boldsymbol{n}}}}}}}}_{{{{{{{\rm{C}}}}}}}}){{{{{{{\boldsymbol{n}}}}}}}}_{{{{{{{\rm{C}}}}}}}}/\left\vert f\right\vert\). ("Part.'', Eq. (6), blue; “Cell'', Eq. (8), red; “Mod. Part.'', Eq. (12), cyan; “Mod. Cell'', Eq. (11), yellow).

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The Pearson correlation coefficient between the direction of the active force and the tracer velocity \({{{{{{\mathcal{P}}}}}}}_{{\hat{v}}_{T},\,\hat{f}}\) is similar for all formulations, with a small peak for αeq ≲ α ≲ αturb (Fig. 9b inset). This is the activity regime where the active length scale is comparable to the system size and bend walls form. The positive correlation results from tracers traveling with the bend walls. In the turbulence regime, the dynamics become slightly anti-correlated at a similar scale of \({{{{{{\mathcal{P}}}}}}}_{{\hat{v}}_{T},\hat{f}}\approx -0.05\). This contrasts with \({{{{{{\mathcal{P}}}}}}}_{\left\vert {v}_{T}\right\vert, \left\vert f\right\vert}\) (Fig. 9b), which is decorrelated until the turbulence regime and then rises, indicating that active forcing correlates with the velocity of the tracer. The PCC \({{{{{{\mathcal{P}}}}}}}_{\left\vert {v}_{T}\right\vert, \left\vert f\right\vert}\) of each activity formulation continues to rise for all activities, except particle-carried activity, which falls off once α ≳ α†, due to the algorithm being overcome by density fluctuations. The increasing correlation of the active-force magnitude with the tracer velocity is expected in the turbulence regime, as the active force becomes the primary driver of the tracer particle motion. This correlation analysis shows that the modulated methods do not produce density gradients that dominate over direct active forcing.

Net active forcing

Having characterized the fluctuations associated with each of the definitions of the local activity, it is worth returning to the definitions to better understand how the various activity formulations produce these variations. Since the particle-carried activity formulation establishes a linear relationship between the cell’s activity and particle density, \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{P}}}}}}}}={\rho }_{C}\alpha\), and since the total number of particles is conserved throughout the simulation, the net activity Fα = ∑CαC is constant. Comparing the net activity with the net activity for the homogeneous system, \({F}_{\alpha }^{H}={\sum }_{C}{\alpha }_{C}({N}_{{{{{{{\rm{av}}}}}}}})\) (i.e., the active force input for average density), acts as a measure of energy input per collision operation. For particle-carried activity, this is definitionally unity (Fig. 10).

Fig. 10: Modulation of cellular activity results in active force fluctuations.
figure 10

The temporally averaged active force input into the system, Fα, is computed for each formulation of cellular activity and normalized with the active force input at average density \({F}_{\alpha }^{H}\). While the active force from particle-carried activity remains constant, all other methods exhibit some degree of active force fluctuations. ("Part.'', Eq. (6), blue; “Cell'', Eq. (8), red; “Mod. Part.'', Eq. (12), cyan; “Mod. Cell'', Eq. (11), yellow).

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This is not the case for the other activity formulations, since they are not linear with local density (Fig. 2). While Fig. 10 appears to indicate that the cell-carried activity is also independent of α, there is in fact a marginal deviation from a homogeneous system at the highest activities. This is due to entirely empty cells in the highest activity regime, which results in a small drop in the net active forcing Fα input (Fig. 2; Eq. (8)). In contrast, the modulated activities never reach a net active forcing equivalent to an homogeneous system, even for the smallest activities (Fig. 10). This is because inherent fluctuations within MPCD make it exceedingly unlikely that the system will remain at the average density. In the modulated activity formalism, this causes the net active forcing to be slightly attenuated even in the low activity limit. As the activity increases, density fluctuations increase and the attenuation also rises, reducing the net active forcing. Between the two modulated formulations, \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MP}}}}}}}}\) has a greater loss in net active forcing because the cell activity at average density \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MP}}}}}}}}(\left\langle {\rho }_{C}\right\rangle )\) is much larger than \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MC}}}}}}}}(\left\langle {\rho }_{C}\right\rangle )\) (Fig. 2). Overall, this shows that these alternative formulations for cell activity trade off reduced density fluctuations for an increase in active forcing fluctuations.

Discussion

Active-nematic MPCD offers many advantages for simulating active fluids in complex environments or with suspended mesoscale inclusions, such as polymers and filaments. However, like any coarse-grained approach, AN-MPCD also possesses limitations and an operational regime. Here, we have considered how activity induces density variations in AN-MPCD, as it does in any particle-based approach28,29,30,31,32. Any modifications that can curb density variations will extend the applicability of AN-MPCD for simulations of homogeneous fluids.

This manuscript considered four activity formulations as a function of local density: (i) The original particle-carried activity \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{P}}}}}}}}\), which is proportional to the local density, (ii) a modified cell-carried activity \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{C}}}}}}}}\), which ensures activity is homogeneously applied throughout the system domain, (iii) modulated particle-carried activity \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MP}}}}}}}}\), (iv) and modulated cell-carried activity \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MC}}}}}}}}\). A sigmoidal modulation is chosen to smoothly switch activity on/off. Particle-carried activity models a system in which local active stress is linearly proportional to the density of active elements, such as bacterial suspensions49 or the limit of low ATP concentration in microtubule/kinesin systems57,58. On the other hand, cell-carried activity imposes constant activity, which is the case in the high ATP concentration limit in microtubule/kinesin systems57,58. Finally, the two modulated formulations introduce a negative feedback by effectively decreasing cellular activity whenever the instantaneous local density is too large, which prioritizes homogeneous density as is commonly assumed in hydrodynamic simulations5,15. All four formulations are found to reproduce the theoretically expected scaling laws of active nematic turbulence, albeit with slightly differing turbulence regimes. The modulated activities \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MP}}}}}}}}\) and \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MC}}}}}}}}\) are found to have the widest effective turbulence regime, roughly four times wider than the particle-carried activity \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{P}}}}}}}}\).

All four formulations are found to exhibit density fluctuations, common to active particle based methods, but the new formulations (\({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{C}}}}}}}}\), \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MP}}}}}}}}\), \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{MC}}}}}}}}\)) exhibit reduced variations over the original particle-carried activity \({\alpha }_{{{{{{{\rm{C}}}}}}}}^{{{{{{{\rm{P}}}}}}}}\) formulation. The modulated activities, in particular, are found to have significantly reduced density fluctuations. While empty cells are still present in all formulations at the highest activities, modulated particle-carried activity is found to have the least number of empty MPCD cells with only 1:1000 empty cells, as opposed to the 1:50 in particle-carried activity. Modulated cell-carried activity has around 1:600 empty cells, but is otherwise comparable. To verify the suitability of AN-MPCD for the study of solutes suspended within an active fluid, the density gradients characterized density-induced Fickian fluxes. Within reasonable limits, the modulated activities are found to have comparable density gradients to passive MPCD. Finally, the Pearson correlation coefficients between tracer particle velocities and local density gradients or active forces confirm that density gradients have minimal effect on the velocity of solutes, which are in fact driven by the active forcing, which is vital for future studies to accurately reproduce the results found in experiment and in theoretical studies75. We summarize the strengths, weaknesses, and additional considerations for the four different collision operators in S4 in the Supplementary Material.

Both human crowds76 and quorum sensing bacteria77,78 have been shown to exhibit slowdown effects in dense environments. These result in a local velocity-density profile akin to the modulation function \({{{{{{{\mathcal{S}}}}}}}}_{C}\) applied to the active forcing of modulated activities in this work, suggesting that modulation of active velocity can mitigate density fluctuations in the general particle case. Based on the results reported here, future research studying other active particle systems, such as dry nematics28,73 and active Brownian particles34, should consider the effects of spatial activity modulation. In particular, further studies of how local density affects activity, whether through local fuel concentration or otherwise, would be valuable.

Although simple in concept and implementation, the AN-MPCD algorithm has much potential for simulating hypercomplex active materials79—suspensions of mesoscale solutes embedded in an active background. Both composite materials and activity are ubiquitous in biology. Biopolymer composites, such as cytoskeletal components or cells embedded in extra-cellular polymeric substances exemplify mesoscale active/passive mixtures. While many studies consider active particles in passive environments, AN-MPCD opens a pathway for simulating passive particles dispersed in an active host.