Clara Del Junco

André Estevez-Torres

Ananyo Maitra

Front speed and pattern selection of a propagating chemical front in an active fluid

Spontaneous pattern formation in living systems is driven by reaction-diffusion chemistry and active mechanics. The feedback between chemical and mechanical forces is often essential to robust pattern formation, yet it remains poorly understood in general. In this analytical and numerical paper, we study an experimentally-motivated minimal model of coupling between reaction-diffusion and active matter: a propagating front of an autocatalytic and stress-generating species. In the absence of activity, the front is described by the the well-studied KPP equation. We find that front propagation is maintained even in active systems, with crucial differences: an extensile stress increases the front speed beyond a critical magnitude of the stress, while a contractile stress has no effect on the front speed but can generate a periodic instability in the high-concentration region behind the front. We expect our results to be useful in interpreting pattern formation in active systems with mechano-chemical coupling in vivo and in vitro.

I. INTRODUCTION

The emergence of form in the living embryo involves a series of out-of-equilibrium chemical and physical processes [START_REF] Collinet | Programmed and self-organized flow of information during morphogenesis[END_REF]. Among them, two mechanisms stand out: reaction-diffusion (RD) [START_REF] Cross | Pattern Formation and Dynamics in Nonequilibrium Systems[END_REF][START_REF] Cross | Pattern formation outside of equilibrium[END_REF][START_REF] Mathison | The chemical basis of morphogenesis[END_REF][START_REF] Kondo | Reaction-diffusion model as a framework for understanding biological pattern formation[END_REF][START_REF] Jeremy | Positional information and reaction-diffusion: two big ideas in developmental biology combine[END_REF], which creates well-defined concentration patterns, and active matter (AM), mediated by molecules that convert chemical energy into mechanical forces generating hydrodynamic flows [START_REF] Gross | How active mechanics and regulatory biochemistry combine to form patterns in development[END_REF][START_REF] Marchetti | Hydrodynamics of soft active matter[END_REF][START_REF] Prost | Active gel physics[END_REF][START_REF] Jülicher | Hydrodynamic theory of active matter[END_REF][START_REF] Ramaswamy | The mechanics and statistics of active matter[END_REF][START_REF] Toner | Hydrodynamics and phases of flocks[END_REF][START_REF] Ramaswamy | Active matter[END_REF][START_REF] Lokrshi Prawar Dadhichi | Origins and diagnostics of the nonequilibrium character of active systems[END_REF][START_REF] Solon | Active brownian particles and run-and-tumble particles: A comparative study[END_REF][START_REF] Cates | Motility-induced phase separation[END_REF]. RD dynamics have also been used to model a wide range of situations in biology ranging from spatiotemporal protein dynamics in cells [START_REF] Eliavs | Reaction-diffusion systems for spatio-temporal intracellular protein networks: A beginner's guide with two examples[END_REF] to the migration of a cell monolayer [START_REF] Macarthur | Chapter 2 -from mathematical modeling and machine learning to clinical reality[END_REF][START_REF] Bram | Experimental characterization and computational modelling of two-dimensional cell spreading for skeletal regeneration[END_REF]. In these contexts, the species undergoing RD dynamics may regulate or control active forces in the living system. Indeed, the importance of interactions between chemistry and mechanics in modeling living systems has long been recognized [START_REF] Mathison | The chemical basis of morphogenesis[END_REF], yet RD and AM systems have generally been examined separately, whether theoretically [START_REF] Mathison | The chemical basis of morphogenesis[END_REF][START_REF] Marchetti | Hydrodynamics of soft active matter[END_REF] or in in vivo [START_REF] Kondo | Reaction-diffusion model as a framework for understanding biological pattern formation[END_REF][START_REF] Müller | Differential diffusivity of nodal and lefty underlies a reaction-diffusion patterning system[END_REF] and in vitro [START_REF] Irving R Epstein | An introduction to nonlinear chemical dynamics: oscillations, waves, patterns, and chaos[END_REF][START_REF] Fj Nedelec | Self-organization of microtubules and motors[END_REF] experiments. More recently, however, mechano-chemical couplings have been investigated theoretically [START_REF] Bois | Pattern formation in active fluids[END_REF][START_REF] Kumar | Pulsatory patterns in active fluids[END_REF], demonstrating the ability of such systems to display novel patterns even in the absence of chemical reactions [START_REF] Bois | Pattern formation in active fluids[END_REF]. When combined with chemical reactions, the coupling modifies the patterning behavior of RD systems [START_REF] Bois | Pattern formation in active fluids[END_REF], and can also generate spatiotemporal oscillations [START_REF] Kumar | Pulsatory patterns in active fluids[END_REF]. Their importance for in vivo assays has been established [START_REF] Collinet | Programmed and self-organized flow of information during morphogenesis[END_REF][START_REF] Gross | How active mechanics and regulatory biochemistry combine to form patterns in development[END_REF], and chemical reactions have also been implicated in the active context in spontaneously creating stable droplets of a finite size and formation of protocells [START_REF] Zwicker | Growth and division of active droplets provides a model for protocells[END_REF]. However, how active flows affect RD dynamics in general remains sparsely understood.

To the best of our knowledge, couplings between RD and AM have not yet been engineered in vitro. However, recent experimental developments suggest that mechano-chemical self-organization could be investigated in well-controlled in vitro experiments in the near future [START_REF] Senoussi | Learning from embryo development to engineer selforganizing materials[END_REF]. Such in vitro systems, which we will call reactiondiffusion active matter (RDAM) systems, would be particularly interesting for quantitatively testing theoretical predictions concerning the propagation of a reacting and diffusing activity regulator -a situation that is likely to be ubiquitous in in vivo settings.

In this paper, we present an examination of how active flows affect the propagation of a reacting and diffusing chemical species that controls the strength of activity, using a highly simplified model. We describe the simplest, non-trivial RDAM system that could be reasonably assembled in vitro with current knowledge (Sec. II). It is an RD system generating a traveling front that is coupled to a compressible active fluid. We model it with a dynamical equation for the concentration of the reactant and a constitutive equation for the velocity field of the active fluid [START_REF] Bois | Pattern formation in active fluids[END_REF] (Sec.III). A linear stability analysis of these equations about a homogeneous state shows that there is a spatial instability for a contractile RDAM system (Sec. IV). However, the linear theory falls short of being able to predict the behavior of the system when symmetry is broken by a traveling front. In particular, a linear theory incorrectly predicts that the front speed is insensitive to the active stresses. Moreover, it does not provide a prediction of how the spatial instability propagates or how its selected wavelength is shifted in the presence of the front. Numerical simulations of the two dynamical equations provide the answers to these questions (Sec. V A). The simulations show that, in extensile fluids, fronts propagate faster as the active stress increases and spatial instabilities are never observed. We show that the front speed picked out in simulations is the minimum physically allowed speed, consistent with a marginal stability mechanism that shows up in much simpler systems, but which does not hold in our highly nonlinear model.

In turn, contractile fluids generate fronts whose velocity is independent of the active stress, and they display spatial instabilities whose group velocity is not equal to the speed of the front and whose wavenumber is shifted to smaller values compared to the linear prediction.

II. IN VITRO REACTION-DIFFUSION ACTIVE MATTER

In this section we will describe an RDAM system that could be assembled in vitro with the current experimental state of the art. We will take into account known experimental constraints to motivate the theoretical description undertaken in the rest of the paper. Although a wide array of AM experimental systems exist, we will restrain ourselves to molecular active matter because we believe that it has the highest chance to be combined with molecular RD systems. In this category, extracts of cytoskeletal filaments and motors, either actomyosin or kinesin/microtubule gels [START_REF] Senoussi | Learning from embryo development to engineer selforganizing materials[END_REF][START_REF] Andorfer | From isolated structures to continuous networks: A categorization of cytoskeleton-based motile engineered biological microstructures[END_REF], are particularly suitable. Depending on the situation, these gels are locally contractile or extensile and are composed of polar, nematic or isotropic particles [START_REF] Needleman | Active matter at the interface between materials science and cell biology[END_REF]. A wide variety of dissipative behaviors have been observed in these gels in vitro, such as aster formation [START_REF] Fj Nedelec | Self-organization of microtubules and motors[END_REF], global contractions [START_REF] Bendix | A quantitative analysis of contractility in active cytoskeletal protein networks[END_REF][START_REF] Peter | Active contraction of microtubule networks[END_REF], chaotic flows [START_REF] Sanchez | Spontaneous motion in hierarchically assembled active matter[END_REF], active nematics [START_REF] Sanchez | Spontaneous motion in hierarchically assembled active matter[END_REF] and active film buckling [START_REF] Senoussi | Tunable corrugated patterns in an active nematic sheet[END_REF]. These systems function in aqueous solution, at neutral pH and room temperature.

Chemical systems that generate RD patterns have historically relied on strongly acidic reactions [START_REF] Irving R Epstein | An introduction to nonlinear chemical dynamics: oscillations, waves, patterns, and chaos[END_REF] which are incompatible with cytoskeletal extracts. However, in the last 15 years, biochemical RD systems working at neutral pH have been developed [START_REF] Isalan | Engineering gene networks to emulate drosophila embryonic pattern formation[END_REF][START_REF] Loose | Spatial regulators for bacterial cell division self-organize into surface waves in vitro[END_REF][START_REF] Steven M Chirieleison | Pattern transformation with dna circuits[END_REF][START_REF] Padirac | Spatial waves in synthetic biochemical networks[END_REF][START_REF] Tayar | Propagating gene expression fronts in a one-dimensional coupled system of artificial cells[END_REF]. Among them, DNA-only [START_REF] Steven M Chirieleison | Pattern transformation with dna circuits[END_REF] and DNA/enzyme [START_REF] Padirac | Spatial waves in synthetic biochemical networks[END_REF] reaction schemes are particularly interesting because the modularity of the DNA hybridization reaction facilitates the engineering of mechano-chemical couplings with active fluids. In particular, one may envision that the single-stranded (ssDNA) species A produced by these reactions dimerizes DNAmotor conjugates [START_REF] Adam Jm Wollman | Transport and self-organization across different length scales powered by motor proteins and programmed by dna[END_REF][START_REF] Tayar | Active liquid crystals powered by forcesensing dna-motor clusters[END_REF], or releases them from a reservoir [START_REF] Vyborna | Dna-controlled spatiotemporal patterning of a cytoskeletal active gel[END_REF]. Indeed, a number of active fluids become active only in the presence of dimers of motors [START_REF] Fj Nedelec | Self-organization of microtubules and motors[END_REF][START_REF] Sanchez | Spontaneous motion in hierarchically assembled active matter[END_REF] and thus, with this scheme, active stresses would be generated only in the presence of species A.

With such a design, we will consider an autocatalytic reaction of ssDNA species A, which has been reported to generate RD fronts propagating at constant velocity in a thin microchannel [START_REF] Anton S Zadorin | Synthesis of programmable reaction-diffusion fronts using dna catalyzers[END_REF] and that is compatible with kinesin-microtubule active matter [START_REF] Senoussi | Programmed mechano-chemical coupling in reaction-diffusion active matter[END_REF]. In the following, we will suppose that this autocatalytic reaction takes place in a solution that is in contact with a thin film of an active fluid composed of filaments and motor-DNA conjugates. We note that active thin films, as well as motor-DNA conjugates, have been obtained both for myosin [START_REF] Kumar | Tunable structure and dynamics of active liquid crystals[END_REF][START_REF] Hariadi | Mechanical coordination in motor ensembles revealed using engineered artificial myosin filaments[END_REF] and kinesin [START_REF] Sanchez | Spontaneous motion in hierarchically assembled active matter[END_REF][START_REF] Adam Jm Wollman | Transport and self-organization across different length scales powered by motor proteins and programmed by dna[END_REF][START_REF] Martínez-Prat | Selection mechanism at the onset of active turbulence[END_REF][START_REF] Furuta | Measuring collective transport by defined numbers of processive and nonprocessive kinesin motors[END_REF][START_REF] Sato | Micrometer-sized molecular robot changes its shape in response to signal molecules[END_REF]. Such a thin film of DNA-responsive active fluid could be obtained at a water-oil interface in the presence of a depletion agent, such as polyethyleneglycol, that pushes the filaments against that interface [START_REF] Andorfer | From isolated structures to continuous networks: A categorization of cytoskeleton-based motile engineered biological microstructures[END_REF][START_REF] Kumar | Tunable structure and dynamics of active liquid crystals[END_REF].

III. ACTIVE TRAVELING FRONT MODEL

In the situation just described, the thin film of active fluid can exchange both momentum and matter (i.e., the filaments can diffuse into the fluid above) with the solution where DNA/enzyme reactions take place. At a given point in space, as the concentration of A increases in the bulk solution, motor dimers will form in the film and will generate active flows (Fig. 1). In order to model this 3-variable (DNA A, motors M and filaments F), biphasic system as a 1-variable monophasic system we will make two sets of assumptions. First, we will suppose that the time-scales of motor-motor dimerization via DNA hybridization, τ + M,A , of motor-motor dissociation, τ - M,A , of motor binding to filaments, τ + M,F , and of motor-filament dissociation, τ - M,F , are fast compared with the time-scale of DNA autocatalysis, τ A . Then, one can model such a system with a single average species C, of concentration c, that could be seen as the concentration of motor dimers bound to filaments that undergo an autocatalytic reaction (referred to as autocatalytic motors in the following). Note that the effective diffusion coefficient D of this species is an average of the diffusion coefficient of free A, D A , and of A bound to M, D M , weighted by the time spent in each state [START_REF] Anton S Zadorin | Synthesis of programmable reaction-diffusion fronts using dna catalyzers[END_REF]. Second, we will suppose that the thickness of both the active film, w a , and the DNA/enzyme solution, w d , are small enough such that the corresponding diffusion time τ D = (max(w a , w d )) 2 /2D A τ A , with D A the diffusion coefficient of A. As C is advected by active flows in the thin active film, this transport may in turn influence the concentration of C in the bulk by the rapid diffusion of C across both phases.

To estimate τ + M,A , we suppose that motor-DNA conjugates dimerize through DNA hybridization with second order kinetics. Thus τ + M,A = 1/(κ + M,A M 0 ), where we take κ + M,A ∼ 10 4 -10 6 M -1 s -1 as a typical value of DNA association rate [START_REF] Estevez-Torres | Fourier transform to analyse reaction-diffusion dynamics in a microsystem[END_REF] and M 0 ∼ 1 -100 nM the typical concentration of kinesin in active matter experiments [START_REF] Senoussi | Tunable corrugated patterns in an active nematic sheet[END_REF]. τ - M,A = 1/κ - M,A , where the dissociation rate of a DNA double strand κ - M,A is controlled by its length and can vary in the range κ - M,A ∼ 10 -2 -10 -5 s -1 . τ + M,F = 1/κ + M,F and we take κ + M,F = 5 s -1 from Leduc et al. [START_REF] Leduc | Cooperative extraction of membrane nanotubes by molecular motors[END_REF]. τ - M,A = 1/κ - M,F , and we calculate κ - M,F = v/ r , where v is the velocity of kinesin along microtubules and r the run length before detachment. Furuta et al. [START_REF] Furuta | Measuring collective transport by defined numbers of processive and nonprocessive kinesin motors[END_REF] measured v and r for DNA-promoted kinesin clusters with different numbers of kinesins and found v = 800 nm/min and r = 2 × 10 3 , 4 × 10 3 and 15 × 10 3 , respectively for 1-, 2-or 4-kinesin clusters. τ A has been measured by Zadorin et al. [START_REF] Anton S Zadorin | Synthesis of programmable reaction-diffusion fronts using dna catalyzers[END_REF] to be in the range 10 2 -10 4 s. Finally D A ≈ D M = 10 -10 m 2 s -1 and

A 2A x y w a w d DNA-conjugated motor ssDNA A Filament [A] = 0 [A] = 0 ζ > 0 ζ = 0 FIG. 1.
Scheme of a plausible experimental reaction-diffusion active matter system that can be described using Eqs. ( 1)- [START_REF] Cross | Pattern formation outside of equilibrium[END_REF]. A chamber is filled with a solution containing an RD and an AM subsystem. The RD subsystem is an autocatalytic DNA/enzyme reaction of the form A → 2A, where A is a ssDNA. The AM subsystem is made of filaments and DNAconjugated motors. Due to depletion forces, the AM subsystem forms a film of height wa (segmented pattern) above the RD solution of height w d . A small quantity of A is introduced on the left side of the chamber (dark gray). In the absence of activity (i.e.

[ATP] = 0 and ζ = 0), A is expected to generate a traveling front of constant velocity [START_REF] Anton S Zadorin | Synthesis of programmable reaction-diffusion fronts using dna catalyzers[END_REF]. In the presence of ATP, A will diffuse on the active region, generating motor dimers that will move along the filaments (gray arrows) generating a flow of filaments (black arrows) and thus induce ζ > 0. At the right side of the front, [A] = 0, the motors are monomeric and thus they move along the filaments without generating flow ζ = 0.

w a and w d are in the range 10 -100 µm. We thus get τ + M,A ∼ 10 -10 5 s, τ - M,A ∼ 10 2 -10 5 s, τ + M,F ∼ 0.2 s, τ - M,F ∼ 2 -20 s, τ A ∼ 10 2 -10 4 s and τ D ∼ 1 -100 s. As a result, provided that the motor-DNA system is designed such that M 0 and κ - M,A are on the upper side of the aforementioned ranges, τ A is larger than all the other characteristic times as required by our model.

Beyond the specific and explicit in vitro scenario described here, the RDAM model we will construct is relevant to a wide range of in vivo systems. Indeed, our theory describes all active thin films of uniform thickness h, where the thickness of the film is maintained by the exchange of material with the surrounding medium [START_REF] Salbreux | Hydrodynamics of cellular cortical flows and the formation of contractile rings[END_REF][START_REF] Maitra | Stable long-range uniaxial order in active fluids at two-dimensional interfaces[END_REF], and in which activity is controlled by an activity regulator C which in the absence of active flows would have the following RD dynamics: an unstable lower concentration state of the regulator is changed to a higher concentration state at a front moving at a well-defined speed. Stated in this general form, our theory may be relevant for understanding the propagation of a cell layer in a wound-healing assay [START_REF] Poujade | Collective migration of an epithelial monolayer in response to a model wound[END_REF], where C models the cells or for the dynamics of a chemical active stress regulator in the cellular cortex.

Having established the experimental relevance of an RD system coupled to an active flow, we now construct a model for understanding two key features of such a system: i. whether active flows enhance or inhibit the speed of the RD front and ii. whether the profile of the component C behind the front remains homogeneous or forms a patterned state. For this, we consider a system which is perfectly homogeneous in ŷ and has a moving front of C along x at which a low (zero) concentration state of C changes to a high (non-zero) concentration state. We take c to be the concentration of the species C and v ≡ v x x to be the flow velocity.

With these assumptions, the general equation of motion for the concentration field is

∂ t c = D∂ xx c -∂ x (vc) + f (c) (1)
where D denotes the diffusivity, the second term on the R.H.S. accounts for the advection of the concentration by the flow and f (c) is a generic reaction term with the following properties: i. it destabilizes a steady-state with c ss = 0 and ii. it saturates at finite value c 0 . For most of the paper, we will use a simple form for f (c):

f (c) = rc(1 -c/c 0 ) (2)
but we have checked that our results do not depend qualitatively on this particular form. Note that, in the absence of the advection by the velocity field, (1) constitutes the well-known equation for a traveling wave introduced independently by Fisher [START_REF] Fisher | The wave of advance of advantageous genes[END_REF] and Kolmogorov, Petrovsky, and Piskunov [START_REF] Kolmogorov | A study of the diffusion equation with increase in the amount of substance[END_REF], which we refer to as the KPP equation 1 . The absence of activity can be experimentally realised in two ways (Fig. 1). In the absence of ATP, A can still bind to the DNA-conjugated motors, and thus the average species C will still have the diffusion coefficient D. In the absence of motors, instead, the relevant diffusion coefficient would be D A .

The overdamped equation of motion for the velocity field, relevant for slow flows encountered in biological and biomimetic systems, has the form

(γ -η∂ xx )v = ∂ x [ζ(c)] (3)
where γ is a friction coefficient that accounts for relative motion with respect to the surrounding medium, η is the viscosity of the active layer and ζ(c) is a stress that depends on the regulator concentration. This stress may have a passive component arising from an equation of state for c, but we take it to be primarily active with 1 We choose not to further promote Ronald Fisher's name due to the racist, ableist, and otherwise supremacist and discriminatory views that he championed in his life and work. He published this equation in the journal Annals of Eugenics (now Annals of Human Genetics) [START_REF] Fisher | The wave of advance of advantageous genes[END_REF]. We acknowledge that apart from this equation, many of the tools that we use in this paper-even concepts as common as the standard deviation-were developed by Fisher and colleagues of his with similar views, such as Karl Pearson.

We strive and encourage the reader to think critically about the motivations behind their development. For recent discussions in popular fora, see for example [START_REF] Levy | Eugenics and the ethics of statistical analysis[END_REF][START_REF] Evans | RA Fisher and the science of hatred[END_REF][START_REF] Gelman | RA Fisher and the science of hatred[END_REF]. For scholarly studies, see for example Mackenzie's history/social studies of science work [START_REF] Mackenzie | Statistical theory and social interests: A case-study[END_REF][START_REF] Mackenzie | Statistics in Britain, 1865-1930: The Social Construction of Scientific Knowledge[END_REF]] and a disability and literary studies perspective [START_REF] Lennard | Introduction: Normality, power and culture[END_REF].

ζ(c) > 0 implying a contractile system and ζ(c) < 0 an extensile system. For much of this paper, we will assume a form for ζ(c): ζ(c) = αc, where α is a constant. However, we checked that our results do not depend qualitatively on the form of ζ by also considering two alternate forms of the stress coupling: a quadratic form, ζ c = αc 2 /c 0 , and a saturating form ζ(c) = αc/(1 + c/c 0 ) (see appendix B and Fig. 7). Eq. ( 1) and ( 3) fully characterise the effectively one-dimensional RDAM system.

To facilitate numerical calculations, we now nondimensionalize ( 1) and ( 3) by rescaling x → x/ , t → t/τ and c → c/c 0 where = η/γ is the hydrodynamic correlation length which is proportional to the thickness of the active layer and τ = 1/r is the zero-wavenumber inverse relaxation (growth) rate of concentration fluctuations. The rescaled equations of motion are To understand the behavior of this model, we first consider the linear stability of a state with a homogeneous concentration implied by ( 4) and ( 5), and then numerically analyze how activity modifies the propagation of a concentration front into a state with c = 0.

∂ t c = 1 Da ∂ xx c -∂ x (vc) + c(1 -c) (4)
(1 -∂ xx )v = Ω(c)∂ x c. (5

IV. LINEAR THEORY

We start by examining the linear stability of a state at zero average concentration i.e., c ss = 0, with boundary conditions c(±∞) = 0, v(±∞) = 0. Linearizing (4) and ( 5) about c ss = 0 and Fourier transforming in space and time, we get

-iσc q,σ = 1 - 1 Da q 2 c q,σ (6)
where σ and q are the (non-dimensional) frequency and wavenumber of the perturbation respectively. The dispersion relation is σ = i(1q 2 /Da) which implies that the amplitude of a small wavenumber fluctuation grows, destabilizing the c ss = 0 state. Next, linearizing about

c ss = 1 (c(±∞) = 1, v(±∞) = 0) and defining δc = c -1 and Ω| c=1 ≡ Ω 1 , we get -iσδc q,σ = -1 + 1 Da q 2 - Ω 1 q 2 1 + q 2 δc q,σ . (7)
This implies that a state with c ss = 1 is stable in the long wavlength q → 0 limit where σ = -i. Eq. ( 7), however, suggests an activity-induced nonhydrodynamic, finite wavenumber instability of the homogeneous state for wavenumbers in the range

|q| ∈    -[1 + Da(1 -Ω 1 )] -[1 + Da(1 -Ω 1 )] 2 -4Da √ 2 , -[1 + Da(1 -Ω 1 )] + [1 + Da(1 -Ω 1 )] 2 -4Da √ 2    (8)
when

Ω 1 > 1 + 2 √ Da + Da Da = Ω c (9)
(or Pe 1 > (1 + √ Da) 2 , where Pe 1 ≡ Pe| c=1 ), as sketched in Fig. 2. Since this requires Ω 1 > 0 it can only be achieved in contractile systems. The maximum growth rate is at the wavenumber

q c = -1 + √ Ω 1 Da = -1 + √ Pe 1 .
The instability starts at finite wavenumber in our model, in contrast to the zero wavenumber instability predicted in [START_REF] Bois | Pattern formation in active fluids[END_REF], due to the reaction term which damps concentration fluctuations at zero wavenumber. 2 2 A note of caution: a prediction of a finite wavenumber instability within a hydrodynamic theory should not be taken to imply

We now consider the case where a front moving at a constant speed k interpolates between these two states, such that c(-∞) = 1 and c(∞) = 0, and v(±∞) = 0. The steady-state front must then be described by a function c(ξ), where ξ = xkt which, in the absence of that all systems with the same symmetry, and the same sign of activity, will display this instability, in contrast to prediction for a zero-wavenumber instability. Whether the growth rate turns positive at a non-zero wavenumber depends not only on the lowest order in wavenumber terms retained within the hydrodynamic theory but terms at all orders in wavenumbers that are not. Such terms encode system-specific details and can eliminate a finite wavenumber instability even when a theory retaining only the lowest order in wavenumber terms suggests its presence.

unstable values of q FIG. 2. Growth rate σ of wavenumber q at different values of the active stress parameter Ω1 = Ω|c=1, from linear stability analysis of ( 4) and ( 5) around c = 1 with a linear stress term ζ(c) = αc. The black dashed line is at σ = 0. For contractile stresses above a critical value of Ω1, given in ( 9), the reaction-diffusion active matter (RDAM) system is unstable to perturbations in a finite range of wavenumbers. Extensile stresses (negative Ω) stabilize the system. In this figure, Da = 0.4.

flow-advection due to activity, solves the equation

0 = 1 Da ∂ ξξ c + k∂ ξ c + c(1 -c). ( 10
)
The speed of the c = 1 front in the absence of activity is expected to be, in dimensional form, k = 2(Dr) 1/2 , using a marginal stability argument originally due to [START_REF] Dee | Propagating pattern selection[END_REF]. Contractile or extensile activity qualitatively modifies the KPP concentration dynamics via the advection by the hydrodynamic velocity. Formally solving [START_REF] Kondo | Reaction-diffusion model as a framework for understanding biological pattern formation[END_REF] in terms of the concentration field leads to an integro-differential equation for the concentration which is not easy to deal with analytically. We will therefore examine effect of advection due to active flow on the front propagation numerically. However, we can first gain some qualitative insight by assuming that v ∂ xx v in [START_REF] Kondo | Reaction-diffusion model as a framework for understanding biological pattern formation[END_REF], which is generically the case when → 0, and replace the velocity field in [START_REF] Mathison | The chemical basis of morphogenesis[END_REF]

with v = Ω(c)∂ x c to obtain 0 = 1 Da -cΩ ∂ ξξ c + [k -(cΩ + Ω)∂ ξ c] ∂ ξ c + c(1 -c) (11)
where the prime denotes differentiation with respect to c. Therefore, the coupling to an active species leads to two distinct modifications of the usual KPP equation: i. it reduces the (non-dimensional) diffusivity at nonzero concentrations via the term cΩ and ii. it leads to a new nonlinear term (cΩ + Ω)∂ ξ c in the equation of motion. The former implies that a c = 1 concentration profile behind the front cannot be homogeneous for large values of Ω, as seen in the linear stability analysis. Note, however, that linearizing [START_REF] Ramaswamy | The mechanics and statistics of active matter[END_REF] suggests an incorrect value for the threshold of the instability of the c ss = 1 state discussed earlier, the range of unstable wavenumbers, and the wavenumber at which the growth rate of the instability is maximum. It predicts an instability for Ω 1 = 1/Da and, since there is no stabilizing mechanism at large wavenumbers, no fastest growing mode. The reason for this is that we assumed the hydrodynamic correlation length to be 0 which amounts to replacing the factor 1/(1 + q 2 ) in the last term of the R.H.S. in ( 7) by 1. If we expand 1/(1 + q 2 ) in a binomial series for q → 0, we see that this approximation ignores even linear terms at higher wavenumbers. Therefore, this implies that [START_REF] Ramaswamy | The mechanics and statistics of active matter[END_REF] becomes qualitatively different from the full dynamics when Ω 1/Da and cannot be used to obtain qualitative information about it beyond this point.

If instead of setting → 0 and ignoring the ∂ xx v term in [START_REF] Kondo | Reaction-diffusion model as a framework for understanding biological pattern formation[END_REF], we formally invert the differential operator

(1 -∂ xx ) -1 ≡ ∞ n=0 ∂ 2n
x , we find that the non-local character of the full velocity field leads to linear and nonlinear higher order in gradient terms in the concentration equation which should lead to wavenumber selection behind the front when the homogeneous state is unstable.

Finally, even in the regime where [START_REF] Ramaswamy | The mechanics and statistics of active matter[END_REF] is predictive, the nonlinear modification of diffusivity and speed implies that results regarding the speed of the front derived using marginal stability analysis for KPP equations cannot be carried over to this case [START_REF] Dee | Propagating pattern selection[END_REF][START_REF] Wim Van Saarloos | Front propagation into unstable states: Marginal stability as a dynamical mechanism for velocity selection[END_REF][START_REF] Wim Van Saarloos | Front propagation into unstable states[END_REF]. In particular, if [START_REF] Ramaswamy | The mechanics and statistics of active matter[END_REF] is linearized around c = 0, it simply reduces to the usual KPP equation [START_REF] Jülicher | Hydrodynamic theory of active matter[END_REF] which would predict that the front speed would be unaffected by activity Ω. This turns out to be incorrect, as we explicitly demonstrate in the next section by numerically simulating (4) and (5).

V. NUMERICAL RESULTS

To explore the propagation of a concentration front in the RDAM system we numerically simulated (4) and [START_REF] Kondo | Reaction-diffusion model as a framework for understanding biological pattern formation[END_REF]. In Fig. 3 we show example trajectories of these simulations. In this section we consider only stress linear in c, i.e. ζ(c) = αc, so that Ω is independent of c and we drop the subscript on Ω 1 . Further examples are shown in the appendix Fig. 7 with alternative forms of the stress term ζ(c) and the reaction term f (c). We simulated the system using custom python code, which is available along with the analysis scripts used in this paper at https://github.com/cdeljunco/rdam. The coupled equations were integrated in real space using an implicit Euler scheme. Discretization parameters (grid size dx, time step dt) were chosen to be small enough that the simulations were numerically stable and results did not depend on them. Box length L and total time T were chosen to be large enough that the simulation results reached their asymptotic values. Dirichlet boundary conditions were imposed on c and v with c(0) = 1, c(L) = 0 and v(0) = v(L) = 0, implying ∂c x (0) = ∂c x (L) = 0 due to equation [START_REF] Kondo | Reaction-diffusion model as a framework for understanding biological pattern formation[END_REF].Da= rη Dγ was chosen to match experimentally realizable conditions with r = τ -1 A = 10 -2 -10 -4 s -1 , the rate of the DNA/enzyme autocatalytic reaction and D = D A = 10 -10 m 2 s -1 , the diffusion coefficient of a 10-mer single-stranded DNA [START_REF] Anton S Zadorin | Synthesis of programmable reaction-diffusion fronts using dna catalyzers[END_REF]. For a thin film of thickness h, η/γ ∼ h 2 , yielding Da = 1 -10 4)-( 5)) studied in this article. This model of mechanochemical coupling may represent, for example, a simplified 1D version of an experiment in a thin film where a reacting and diffusing chemical species A interacts with stress-generating molecules (for instance, molecular motors walking on filamentous proteins) such that the magnitude of the stress increases with increasing concentration of A. Positive values of Ω model contractile stresses, while negative values model extensile stresses. Similar to Ref. [START_REF] Bois | Pattern formation in active fluids[END_REF], we find that a periodic instability develops behind the front above a critical value of contractile stresses. Moreover, we note several other interesting features which we explore in this paper. First, there is an asymmetric effect of stresses on the speed of the traveling front, which is increased by extensile stresses but unaffected by contractile stresses. Second, the instability is modified by the boundary condition imposed by the front in three ways: the critical value of Ω is shifted to a higher value compared to an infinite system; there is a gap between the front and the instability which closes with increasing Ω; and the instability selects a single wavelength from the spectrum of unstable wavelengths in the ideal system. Simulation parameters for all simulations in this figure: L = 400, T = 100, dx = 0.01, dt = 0.1, Da = 0.4. h = 100 µm. We chose Da ranging from 0.2 to 0.8 in the following. Finally, the stress parameter Ω was varied in the range -30 to 12 to probe different regimes of contractile and extensile stresses. Specific parameter values are stated along with the results in what follows.

Fig. 3 summarizes our central findings. It shows that with contractile stresses the front speed does not change significantly and, above a critical value of Ω, an instability appears behind the front where the concentration is high, as predicted from the linear theory. In contrast, in the extensile case, the selected front speed k * increases as a function of Ω but no spatial instability is observed. We now examine these results in more detail.

A. Front speed selection in the presence of activity

In this section, we examine how activity modifies the speeds of the concentration fronts. We defined the front location as the rightmost point where c = 0.5 (when the contractile instability sets in, there can be points behind the front with c = 0.5). We computed the location of the front from the simulations in Fig. 3 and found that it is a linear function of time after an initial transient resulting from the step function initial conditions that we used. This is because the speed of a KPP front and its shape are related; specifically, linearizing (10) about c = 0 and assuming that c(x, t) is a traveling wave with the exponential form c(x, t) ∝ exp(-a(xkt)) at the leading edge, yields a dispersion relation which predicts that the front speed k decreases with increasing a, saturating at k = 2Dr -1/2 for a ≥ 1 [START_REF] Murray | Mathematical Biology I[END_REF]. As a result, simulations begun from an infinitely steep front must accelerate to their asymptotic speed. We therefore calculated the front speed from a linear fit of the front position vs. time at late times after the speed had stopped changing. We verified our results by checking that we obtained the same front speed in simulations initialized with a shallow initial profile of the form c(x, 0) = tanh((xx 0 )/m) with m chosen to be large enough that the front speed converges to its final value from above. Additionally, we checked that our choices of lattice spacing (dx = 0.5) and time step (dt = 0.1) were small enough that further reducing them had a negligible effect on the calculated front speed.

We first consider the effect of extensile activity on the front speed. In our reduced model [START_REF] Ramaswamy | The mechanics and statistics of active matter[END_REF], if Ω is independent of c, the activity nonlinearly enhances the ∂ ξξ term and reduces the ∂ ξ term compared to the standard KPP equation, making it quite different from the standard KPP equation [START_REF] Jülicher | Hydrodynamic theory of active matter[END_REF]. It follows from a simple physical argument that restricts solutions to have non-negative values of c that a KPP front can adopt any speed above a minimum allowed value [START_REF] Murray | Mathematical Biology I[END_REF], but it has been shown using marginal stability analysis that in fact the front asymptotically picks out this minimum speed [START_REF] Dee | Propagating pattern selection[END_REF]. While a marginal stability argument is not expected to yield the front speed in this case, we hypothesize that the selected front speed in full simulations (k * ) will still be given by the minimal allowed value of k, k min , for which c is non-negative in simulations of [START_REF] Ramaswamy | The mechanics and statistics of active matter[END_REF]. We numerically integrated [START_REF] Ramaswamy | The mechanics and statistics of active matter[END_REF] for each value of Ω using Matlab's ode23s function for a range of values of k (code online at https://github.com/cdeljunco/rdam). The minimum allowed value of the front speed, k min , was identified as the smallest value of k for which c ≥ 0 for all ξ. The k min obtained via this procedure is an increasing function of Ω (in contrast to the prediction from a naive linearization of ( 11) about the c = 0 state [START_REF] Dee | Propagating pattern selection[END_REF]). This observation can be heuristically accounted for by the following argument. For [START_REF] Jülicher | Hydrodynamic theory of active matter[END_REF], linearizing about c = 0 shows that, due to the constraint that c ≥ 0, the only possible trajectories have k ≥ 2(Dr) 1/2 [START_REF] Murray | Mathematical Biology I[END_REF] (which is also the speed of the front [START_REF] Dee | Propagating pattern selection[END_REF]). Now, since c must always be positive and ∂ ξ c is always negative in permitted trajectories where the front moves from left to right, in [START_REF] Ramaswamy | The mechanics and statistics of active matter[END_REF] a negative value of Ω (corresponding to extensile stresses) increases the diffusivity (D eff = 1/Da -cΩ). At the same time, the part of k eff (which is the coefficient of ∂ ξ c in [START_REF] Ramaswamy | The mechanics and statistics of active matter[END_REF]) that depends on ∂ ξ c is -Ω∂ ξ c and therefore decreases. Since k eff is constrained to be greater than D eff , and since D eff increases and -Ω∂ ξ c decreases as Ω becomes more negative (i.e., for increasing extensile stress), the value of k must increase to compensate. Thus, the allowed values of front speed shift upwards.

We now compare this result with the full numerical simulation to test our hypothesis. Fig. 4a shows that the front speed begins to increase in extensile simulations for Ω ≈ -5 (for Da = 0.4). In Fig. 5 we compare k * to k min for three values of Da and 0 < |Ω| < 25. Fig. 5b shows that k min is within 5% of k * for the largest value of Da that we looked at (Da = 0.8), and is within 1% for the smallest value Da = 0.2. This is a significant finding for two reasons: i. Eq. ( 11) was obtained within an → 0 approximation and therefore ignored both linear and nonlinear terms of higher order in gradients. Despite this, the reduced model predicts the front speed reasonably accurately. ii. This demonstrates that even in the presence of activity, which makes the effective concentration equation non-local due to the non-local dependence of the velocity field on the concentration, the selected speed is still given by the minimum allowed speed as had been predicted by [START_REF] Dee | Propagating pattern selection[END_REF][START_REF] Wim Van Saarloos | Front propagation into unstable states: Marginal stability as a dynamical mechanism for velocity selection[END_REF] in the context of much simpler models. The selected speed in both the full simulation and from [START_REF] Ramaswamy | The mechanics and statistics of active matter[END_REF] remains equal to the speed in the absence of activity up to a critical value |Ω * | and then starts to increase with |Ω| as shown in Fig. 5a. Fig. 5a also demonstrates that Ω * decreases with increasing Da. This feature may also be heuristically understood from [START_REF] Ramaswamy | The mechanics and statistics of active matter[END_REF]: when cDa|Ω| 1, the effective diffusivity in ( 11) is effectively controlled by 1/Da and the speed is equal to the one in the absence of activity. However, as Da increases, |Ω| ∼ 1/Da at smaller values of Ω implying that the front speed is controlled by activity for |Ω| > |Ω * | that decreases with Da.

Comparing the results in Fig. 5 with the values of the experimental parameters summarized in section III allows us to evaluate the magnitude of the increase in the front velocity expected in the experiment sketched in In these conditions, from Fig. 5, we expect a change in the front velocity of about 20% in the presence of activity, which could be reasonably measured experimentally.

We now consider the effect of contractile active stress on the front propagation. As discussed earlier, ( 11) is only expected to be predictive for Ω < 1/Da. We find that, in this regime (i.e., when [START_REF] Ramaswamy | The mechanics and statistics of active matter[END_REF] does not imply an instability of the high concentration state), [START_REF] Ramaswamy | The mechanics and statistics of active matter[END_REF] implies that the minimum allowed value of k is essentially the same as the one for Ω = 0. This is heuristically understandable: even for the extensile system, the front speed was only controlled by activity, and increased with it, when |Ω| 1/Da. Therefore, since [START_REF] Ramaswamy | The mechanics and statistics of active matter[END_REF] is only valid for Ω < 1/Da, we don't expect the front speed to decrease with contractility at least when the high concentration front is not itself unstable. Our results from the full numerical simulations confirm that the front speed is not modified when the active stress is contractile, as shown in Fig. 4. Surprisingly, we find that the front speed remains independent of Ω even at larger values of Ω when the concentration and velocity profiles behind the front develop a periodic pattern, which we discuss in the next section.

B. Patterns behind the front with contractile stresses

We now consider the dynamics at large Ω when the homogeneous profile behind the front is destabilized and a periodic pattern is formed. We measure Ω c -the minimum value of Ω for which a periodic pattern emergesand q c -the wavenumber of the periodic pattern -from the simulation of the full model. We checked that in simulations with periodic boundary conditions Ω c and q c are consistent with linear stability analysis predictions (see Appendix C for methods and results).

In simulations with a front, there is a small peak in the concentration just behind the front, which we identified using a standard peak-finding algorithm as the peak with the largest value of x. We call the location of this peak x f as it tracks the location of the front. We define the leading edge of the instability x p1 to be the location of second leftmost peak. We then Fourier transformed the region from x = 0 to x = x f , as shown in Fig. 6a. Because x p1 is a function of time, the region of the Fourier transform changes over time and so do the possible wavenumbers of the instability. We therefore approximated the height of the dominant wavenumber at long times (q c ) by the amplitude h max of the wavenumber with maximum amplitude (q 1 ) at any point in time. We plotted log(h max ) vs. t and fit the linear region of the curves to extract the growth rate σ(q 1 ; Ω). We then fit Ω vs. σ(q 1 ) to a line and extrapolated Ω c as the value of Ω where σ(q 1 ) = 0.

Surprisingly, we find from this analysis that Ω c ≈ (1 + 2 √ Da + Da)/Da as predicted by the linear stability analysis of a homogeneous c ss = 1 state. Next, examining time-evolution of the dominant wavenumber q 1 , we find that it relaxes from below to a constant mean value q c after t ≈ 1000. In Fig. 6b, we plot the mean ± 1 standard deviation of q 1 from t = 1200 -1800 (i.e., q c ± 1 standard deviation), measured at intervals of ∆t sample = 2. The value of q c is within 1% of the value predicted by the linear stability analysis when Ω is just above Ω c , and diverges from it as Ω c increases.

Finally, we measured the lag of the leading edge of the instability behind the front (i.e. the front speed k * minus the periodic pattern's group velocity v g ). After a brief initial transient, the distance ∆x instab = x f -x p1 is a linear function of time. We calculated k * -v g as the slope of the linear fit to t vs ∆x instab . As shown in Fig. 6c, k * -v g goes from a positive value just beyond Ω c to 0. That is, the pattern lags behind the travelling front till a second value Ω * c . Before this value, the front "outruns" the periodic pattern implying that near the travelling edge, a homogeneous region emerges before the patterned state sets in (see supplemental movie). Beyond Ω * c , the pattern "catches up" with the front and for all higher values, moves with the front. The value of Ω * c -Ω c decreases with increasing Da (which depends on the ratio of the reaction coefficient and the bare diffusivity). This implies that in highly diffusive species, the patterns formed due to the contractile instability lag arbitrarily far behind the advancing front and will not be observed in experiments that track the front. Experimentally, microtubule active matter also displays a contractile behavior and could thus be used to test these predictions, either with multiheaded kinesins [START_REF] Torisawa | Spontaneous Formation of a Globally Connected Contractile Network in a Microtubule-Motor System[END_REF], clustered kinesins [START_REF] Senoussi | Programmed mechano-chemical coupling in reaction-diffusion active matter[END_REF][START_REF] Bezia | Active microphase separation in mixtures of microtubules and tip-accumulating molecular motors[END_REF] or dyneins [START_REF] Peter | Active contraction of microtubule networks[END_REF].

VI. CONCLUSION

In this paper we examined the effect of active flows on the propagation of a reacting and diffusing activity regulator species. We demonstrated that neither contractile nor extensile active flows inhibit the propagation of the activity regulator. We further showed that extensile activity (corresponding to negative Ω = Pe/Da in our model) increases the propagation speed of the activity regulator relative to k 0 , the speed in the absence of activity, when the magnitude of Ω exceeds a critical value Ω * , while contractile activity does not modify the front speed. On the other hand, contractile active flows above a critical value Ω c destabilize the homogeneous state behind the front leading to a periodic pattern. For moderate Da and activity, however, a homogeneous state prevails near the front edge as the speed of the pattern lags behind the front speed.

RDAM systems may a wide range of in vivo biological systems. For instance, they may be relevant for the dynamics of activity regulators [START_REF] Bois | Pattern formation in active fluids[END_REF] in the cellular cortex or may describe the dynamics of cell monolayers [START_REF] Macarthur | Chapter 2 -from mathematical modeling and machine learning to clinical reality[END_REF]. They may also be synthesized in vitro. Indeed, we motivated our 1-dimensional, 1-variable model from the state-of-the art in vitro experiments involving a DNA/enzyme autocatalytic RD front coupled with a DNA-responsive kinesin/microtubule active fluid. We believe that the qualitative features predicted by the theory presented here can be observed in in vitro RDAM systems such as the one suggested here. Taken together, our results may prove useful in interpreting pattern formation in RDAM systems in vivo and in vitro that are on the experimental horizon.

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  ) where c, x, t and v are now non-dimensional, and Da = rη/Dγ -the second Damköhler number -and Ω(c) = ∂ c ζ(c)/rη are non-dimensional parameters of the model. Da and Ω express ratios, respectively, of characteristic timescales of reaction to diffusion, and of active transport to reaction. Thus, Ω can also be expressed as Ω = Pe/Da, where Pe is the more familiar Péclet number expressing the ratio of active to diffusive time scales, given in our case by Pe(c) = ∂ c ζ(c)/Dγ.

FIG. 3 .

 3 FIG.3. Representative simulations of the RDAM model ((4)-(5)) studied in this article. This model of mechanochemical coupling may represent, for example, a simplified 1D version of an experiment in a thin film where a reacting and diffusing chemical species A interacts with stress-generating molecules (for instance, molecular motors walking on filamentous proteins) such that the magnitude of the stress increases with increasing concentration of A. Positive values of Ω model contractile stresses, while negative values model extensile stresses. Similar to Ref.[START_REF] Bois | Pattern formation in active fluids[END_REF], we find that a periodic instability develops behind the front above a critical value of contractile stresses. Moreover, we note several other interesting features which we explore in this paper. First, there is an asymmetric effect of stresses on the speed of the traveling front, which is increased by extensile stresses but unaffected by contractile stresses. Second, the instability is modified by the boundary condition imposed by the front in three ways: the critical value of Ω is shifted to a higher value compared to an infinite system; there is a gap between the front and the instability which closes with increasing Ω; and the instability selects a single wavelength from the spectrum of unstable wavelengths in the ideal system. Simulation parameters for all simulations in this figure: L = 400, T = 100, dx = 0.01, dt = 0.1, Da = 0.4.

Fig 1 . 1 D

 11 Taking the thin film approximation η/γ ∼ w 2 d , we have Da = w 2 d A τ A ≈ 2 × (10 -3 -10 -1 ) for a reasonable value of w d = 40 µm. Kinesin/microtubule active fluids generate flows with velocities up to[65] v a = 600 µm/s associated with typical lengths l a = 100 µm, yielding P e = v a l a /D A ≈ 6 and thus Ω = P e/Da ≈ 6×(10 3 -10).

FIG. 4 .FIG. 5 .

 45 FIG.[START_REF] Mathison | The chemical basis of morphogenesis[END_REF]. The effect of extensile and contractile stresses on the selected front speed k * . Extensile active stresses result in an increase in k * above a critical value Ω * of Ω, while contractile stresses have no effect. All y-axes are normalized by k0 = 2Da -1/2 , the front speed at Ω = 0. The line at k * = k0 is a guide for the eye. Simulation parameters for these plots: T = 900, dt = 0.05, dx = 0.1, L = 3000 for Ω > -7 and L = 2800 + 58|Ω| for Ω ≤ -7. L was chosen such that the front would not reach the end of the box before t = 900.

FIG. 6 .

 6 FIG.6. Wavenumber and group velocity of the periodic pattern behind the front. (a) To measure the group velocity vg and the fastest growing wavenumber qc, we defined the location of the front x f as the rightmost peak of the concentration, and the location of first peak of the instability xp1 as the location of the second rightmost peak. (b) qc as a function of Ω, defined as the mean value over the final third of a long simulation of the wavenumber with maximum amplitude in the Fourier transform of the concentration between x = 0 and x = x f . The vertical dashed line is located at Ωc. Parameters for these simulations were: Da=0.4, dt = 0.05, dx = 0.05, L = 6300, T = 1800. (c) Front speed k * minus the group velocity of the periodic pattern vg as a function of Ω at three values of Da (colors). Each simulation data point is obtained from the slope of the distance from the front to the first peak of the instability, ∆x instab = x f -xp1, as a function of time, which is a linear function at long times. The vertical dashed lines are located at Ωc, and the dotted lines at Ω * c , showing decreasing Ω * c -Ωc with increasing Da. Parameters for these simulations were: dt = 0.05, dx = 0.05, L = 500, T = 150.

60 (

 60 front hits right side of box with smaller ) Ω ⇣(c) / c < l a t e x i t s h a 1 _ b a s e 6 4 = " m w w z R j b E p Y

D 8 5

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  z p P z 4 r w 7 H / P W F S e b O Q J / 4 H z + A D y F l H I = < / l a t e x i t > ⇣(c) / 1/(1 + c) < l a t e x i t s h a 1 _ b a s e 6 4 = " p C N G 9

FIG. 7 .FIG. 8 .

 78 FIG. 7. Simulation results with alternative choices of stress and reaction terms. (Top two rows) Representative simulations of the concentration as a function of time in the reaction diffusion active matter (RDAM) model ((4)-(5)), with three alternative forms of the stress term ζ(c) and two forms of the reaction term f (c). The upper left panel is the same as in Fig. 3. Parameters: Da= 0.4, L = 400, T = 100 for all simulations; dx = 0.05 and dt = 0.1 for simulations with ζ(c) ∝ c and ζ(c) ∝ c 2 ; dx = 0.01 and dt = 0.05 for simulations with ζ(c) ∝ c/(1 + c). (Bottom row) Wavenumber with maximum amplitude in the Fourier transform of the concentration behind the front when Ω > Ωc, as a function of time, computed as described in section V B. Parameters: Da= 0.4 for all simulations; T = 1800, L = 6300, dx = 0.05, and dt = 0.05 for simulations with ζ(c) ∝ c; T = 2400, L = 8400, dx = 0.05, and dt = 0.05 for ζ(c) ∝ c 2 ; T = 2400, L = 8400, dx = 0.01, and dt = 0.05 for ζ(c) ∝ c/(1 + c).

ACKNOWLEDGMENTS

We thank Raphaël Voituriez, Suri Vaikuntanathan, Jean-Christophe Galas, Anis Senoussi and Yuliia Vyborna for helpful discussions. This work has been funded by a Natural Sciences and Engineering Research Council of Canada Postgraduate Scholarship -Doctoral (NSERC PGS-D, C.d.J.); by the University of Chicago Materials Research Science and Engineering Center (MRSEC, C.d.J.), which is funded by the National Science Foundation under Award No. DMR-1420709; and by the European Research Council (ERC) under the European's Union Horizon 2020 programme (grant No 770940, A.E.-T.).

Appendix A: Demonstration that RDAM model is applicable both for aligned and isotropic systems

In the main text, we described a simplified RDAM model in which we did not take orientational ordering of the active units into account. However, active matter systems, such as the kinesin-microtubule one described in the main text, are often aligned [START_REF] Senoussi | Tunable corrugated patterns in an active nematic sheet[END_REF]. In this appendix we show that the RDAM model is still applicable in this case as long as the speed of the KPP front is fast compared to the relaxation of the angular fluctuations in the aligned state.

We consider an aligned system (for instance, composed of microtubules or polarised cells) with the degree of alignment denoted by

where S is the degree of order with S = 0 denoting an isotropic organisation and S = 0 denotes an aligned state. With this, the equation for the concentration field is modified to

where D is the bare diffusivity and f (c) is a generic reaction term. The "curvature current" ∝ ζ c is a standard active concentration current in active, aligned systems [START_REF] Marchetti | Hydrodynamics of soft active matter[END_REF]. The equation of the velocity field becomes

where Π is the equilibrium pressure stemming from an equation of state for c, ζc is the isotropic active stress and ζ is the active apolar stress that arises in aligned systems [START_REF] Marchetti | Hydrodynamics of soft active matter[END_REF][START_REF] Simha | Hydrodynamic fluctuations and instabilities in ordered suspen-sions of self-propelled particles[END_REF]. To complete the description of the model, we also need to specify the fluctuations of the order parameter Q. However, we assume that the propagation of the chemical species happens at a time-scale that is much faster than the alignment dynamics and on that timescale, the orderparameter Q can be assumed to be frozen at a constant value. Therefore, we replace Q by its value given by (A1) with S = 0 for an isotropic state and S = 1 for an aligned state. Appendix B: Alternative stress and reaction terms Fig. 7 demonstrates the phenomena that we describe in this paper are also present with several alternative choices for the stress and reaction terms ζ(c) and f (c). Specifically, for two reaction terms and three stress terms, we observe an increased front speed with extensile stresses and an instability behind the front that picks out a single wavelength at long times. Linear stability analysis predicts that instabilities arise when

where µ and β are constants that depend on the choice of ζ(c) and f (c) with values given in Table I. Predicted values of Ω c for each value of µ and β are given in Table II. Fig. 7(a) shows that the location of the instability is shifted in agreement with the linear stability prediction for all choices of f (c) and ζ(c) considered here. While it is not clear from the short simulation in Fig. 7a that the model with saturating stress (ζ(c) ∝ c/(1+c)) picks out a single wavelength, the longer simulations in Fig. 7b show that at long times q 1 converges even in this case. as predicted from a linear stability analysis.

Appendix C: Confirming linear stability analysis results in simulations

In this section we compare our linear analysis in section IV to simulations with periodic boundary conditions. Beginning from a system with a high concentration plus a random initial perturbation, c(x) = 1 + 0.001ν(x) where ν(x) = 0, ν(x)ν(x ) = δ(xx ), we simulated a system with parameters Da = 0.4, dx = 0.01, dt = 0.1, T = 100, and a range of box sizes. Denoting the Fourier transformed concentration as ĉ and assuming that fluctuations in the concentration grow as ĉ(q) ∝ exp(σ(q)t), we measured the growth rate of the fastest growing wavenumber, σ(q c ), as the slope of the best fit line to log(c(q c )) vs. t. We then fit Ω vs. σ(q c ) to a line and extrapolated Ω c as the value of Ω where σ(q c ) = 0. We find that the critical values of Ω c (Fig. 8a) and q c (Fig. 8b -the wavenumber that dominates the spectrum of c at long times), are in good agreement with our linear stability analysis calculations.


Malta

Attard

Livia Holden

Sample Judges Lawyers Experts Beneficiaries

Judges

Out of the three judges who responded to the questionnaire, 2 indicated the upper judiciary as their degree of jurisdiction. Q62 Criminal Tribunal (1), Family Court (1) and 'Other civil law tribunals' were chosen as areas of jurisdiction.

Area of Jurisdiction

Lawyers

Out of the 24 lawyers who responded, 21 indicated their career stage. Mid-career was the most common with 10 (48%), followed by Senior lawyers with 9 (43%) and Junior Lawyers with 2 (10 %). For the areas of law practiced 'Contracts and Obligations' was the most common (15 %, 12), followed by 'Inheritance law' (10%, 8) and 'Property law' (10%, 8), 'Family law' (9%, 7), 'Constitutional law' (8%, 6), 'Immigration law' (6%, 5), 'Refugee and asylum law' (6%, 5), 'Administrative law' (5%, 4), 'Business and commercial law' (5%, 4), 'Criminal law' (4%, 3), 'European law' (4%, 3), and 'other' (4%, 3), 'Banking, bankruptcy, and insolvency law' (3%, 2), 'Labour law' (3%, 2), 'Environmental law' (1%, 1), 'Financial law' (1%, 1), 'Health law' (1%, 1), 'Intellectual and patent law' (1%, 1), 'International and Human rights law' (1%, 1), 'Medical and bio law' (1%, 1), 'Private international law (1%, 1).

Career Stage

Areas of Law

Experts

The most common type of expert selected was 'Translator/Interpreter' on 75% (12), and 'Cultural mediator' on 25% (4).

The most common areas of specialisation were Sub-Saharan Africa on 19% (3), North Africa on 19 % (3), the Middle East on 19% (3), Southern and Central America on 13% (2), Minorities/ Indigenous Population in Europe on 6% (1). Among the 25% who chose Other, 1 indicated Francophonie, 1 Portuguese speaking countries, 1 Maltese law, and 1 the Balkans.

Area of Specialisation

Frequency

Frequency of involvement

To the question of how many experts did they instruct annually, most responding judges and lawyers had instructed experts in less than 10 cases (65 %, 11), followed by 'none of the above' (24%, 4) out of which 3 indicated to have never appointed experts, and 1 indicated to have acted themselves as experts at the SCCRC of Scotland and to appoint anthropologists two to three times a year as pro-bono experts in Malta. To this followed, instructions between 10 and 20 (6%, 1), and between 30 and 50 (6%, 1). 38% (6) of experts provided a total of less than 5 written report, 19% (3) indicated a number of written reports between 20 and 50, 25% (4) indicated a number of written reports between 50 and 100 cases, and 18% (3) indicated 'other'. 25% (4) of experts provided a total of less than 5 oral reports, 6% (1) indicated a number of oral reports between 5 and 10, 13% (2) indicated a number of oral reports between 10 and 20, 6% (1) indicated a number of oral reports between 20 and 50, 13% (2) indicated a number of oral reports between 50 and 100, 37% (6) indicated 'other'. Out of the 6 beneficiaries who responded to the questionnaire, 2 (33 %) said that they had 'always' used cultural expertise, followed by 'prefer not to say' 2 (33 %), followed by 1 (17%) who indicated 'once', followed by 'Other' (17 %, 1) who indicated 'Translator'.

Frequency

Typology of Experts

The most common expert type was 'Native language speakers' (33 %, 8), followed by 'Other' out which 1 indicated 'doctors', then 'country experts' (13%, 3), 'university professors (8%, 2), 'Native lawyers' (8%, 2), 'Community leaders' (4%, 1), and 'Religious leaders' (4%, 1). The 3 (8%) who selected 'university professors' were asked to clarify the disciplines of these professors, to which 1 responded 'anthropology', 1 'Political science', and 1 'Law'.

Expert type

Discipline

% Count

Anthropology 33% 1

Political science 33% 1

Law 33% 1 Total 100% 3

Modalities Appointment of Experts

Lawyers and judges said that experts were selected due to Client/Defendant/Claimant/Applicant's request (14%, 11), followed by the reputation of the expert (14%, 11), then experts facilitate successful legal outcomes (14%, 11), the appointment is advised by the court (11%, 9) time (10%, 8), the law allows the appointment of experts (10%, 8), cost (9%, 7), the court is keen to hear cultural arguments (6%, 5), the court/prosecutor/Home Office have already appointed their expert (6%, 5), expertise can also be used as an out of court settlement (4%, 3), and prefer not to say (1%, 1).

Expert Type

Native Regarding the question of remuneration, the most common response of the experts was 'standard hourly rate' (46 %, 6), followed by 'set price per report' (31 %, 4), then 'voluntary basis' (8%, 1), 'prefer not to say' (8%, 1), 'other' (8%, 1) who indicated 'paid in accordance to attendance to hearing'. The most common response to the question regarding the reuse of cultural expertise was that cultural expert witnessing is applicable to similar cases (38 %, 5), followed by 'prefer not to say' (23 %, 3), 'unique and not repeatable experience' (15%, 2), 'Other' (15%, 2) who indicated the difficulty to find cultural experts linked to the need for re-use of cultural expertise, and 're-use only within the same country/legal field' (8%, 1). Reuse of cultural expertise cultural expert witnessing is applicable to similar cases

Remuneration of experts

Reuse of cultural expertise

Prefer not to say unique and not repeatable experience the difficulty to find cultural experts linked to the need for re-use of cultural expertise re-use only within the same country/legal field

Impact

Components of Impact

The factors most likely to have an impact on their addresses includes the use of first-hand data (19%, 15), reliable sources of contents (19%, 15), stringent conclusions (14%, 11), reputation of the experts (13%, 10), use of statistics (12%, 9), advocacy (6%, 5), remuneration of experts (5%, 4), style (4%, 3), prefer not to say (3%, 2), numerical assessment of risk (3%, 2), other (1%, 1).

Components of impact

Usefulness

The most common response to the question regarding the usefulness of cultural expertise was 'very useful' (35%, 6), followed by 'moderately useful' (18%, 3), 'prefer not to say' (18%, 3), 'extremely useful' (12%, 2), 'slightly useful' 12%, 2), 'not at all useful' (6%, 1).

Usefulness of cultural expertise % Count

Very Most respondents indicated that cultural expertise is most useful in migration law (38%, 8), followed by 'more useful in criminal law than in civil law' (29%, 6), 'more useful in civil law than in criminal law' (10%, 2), useful in all areas of law (10%, 2), in family law (5%, 1), civil and criminal law (5%, 1), asylum, immigration and criminal law (5%, 1).

Usefulness

Competitiveness

Experts thought that their expertise was competitive because of their 'competence' (33%, 8), 'reputation' (33%, 8), 'balance between competence and costs' (21%, 5), and 'convenient hourly quote' (13%, 3).

Areas of law where most useful

Improved access

Database

Regarding the question of the usefulness of a case law database, the most common response was that it would be very useful (65%, 13), followed by being somewhat useful (15%, 3), of no use (10%, 2), prefer not to say (10%, 2). Well over half of the respondents indicated that they would like to contribute to the establishment of a case law database (75%, 12), with the remainder divided between not willing to contribute (13%, 2) not willing to contribute, and prefer not to say (13%, 2).

Reputation of experts

Contribution to law database

% Count

Willing to contribute 75%

Capacity building

Well over the half of the respondents indicated that they thought a teaching program on cultural expertise would be definitely useful (74%, 14), followed by probably useful (16%, 3), might or might not (5%, 1), and prefer not to say (5%, 1).

Usefulness of teaching program on cultural expertise % #

Definitely useful 74% 14 Probably useful 16% 3 Might or might not be useful 5% 1

Prefer not to say 5% 1 Total 100% 19

To the question regarding the interest in a teaching program about cultural expertise, 36% (8) said to know schools, universities and organisations that may be interested in teaching cultural expertise, followed by those interested in teaching cultural expertise themselves (27%, 6), then prefer not to say (27%, 6), then knowing professional organisation being interested in the teaching of cultural expertise (5%, 1), and 'other' (5%, 1).

Usefulness of a program teaching cultural expertise

Less than 10 Never appointe d experts Two to three 10 and 20 30 and 50 ANNUAL CASES Malta Data Summary CULTURAL EXPERTISE IN EUROPE: WHAT IS IT USEFUL FOR? (EURO-EXPERT)

	PI: Livia Holden | Data Collector: Ibtisam Sadegh
	Date of Publication: 01/02/2022

  The most common areas of use of cultural expertise were Refugee and Asylum law 16% (12), Family law 15% (11), immigration and Criminal law both at 11% (8), Contract and Obligation law 8% (6), European law 7% (5), Human Right Law 6% (4), Administrative law and Health law 4% (3), Constitutional, Business and commercial, Labour, Intellectual and Patents, Medical and Biological, all at 3% (2), 1% (1) Inheritance, Property, Private International law and Other.

				Malta Data Summary Malta Data Summary Malta Data Summary
		CULTURAL EXPERTISE IN EUROPE: WHAT IS IT USEFUL FOR? (EURO-EXPERT) CULTURAL EXPERTISE IN EUROPE: WHAT IS IT USEFUL FOR? (EURO-EXPERT)
				PI: Livia Holden | Data Collector: Ibtisam Sadegh PI: Livia Holden | Data Collector: Ibtisam Sadegh
					Date of Publication: 01/02/2022 Date of Publication: 01/02/2022
	Number of Cases Sites The most common site for cultural expertise was 'In court' accounting for 21% (12), followed by Written report Oral report
	NGOs 18% (10), 'Out of Court' 14% (8), 'In detention centers' 11% (6), 'In universities 9% (5), 'In
	hospitals' 9% (5), 'In schools' 7% (4), 'through private consultancies' 7% (4), 'prefer not to say' 4%
	(2), and 'others' 2% (1) who indicated the police.
	Fields of Law Sites	%	% Count	#	Sites
	Contracts and Oblig. 8% In court 21% 12 NGOs 18% 10	6	Fields of Law
	Inheritance Out of court In detention	1% 14% 8 11% 6	1		Less than 5 cases Between 5 and 10
	Less than 5 cases Property centres	1%	1	Less than 5 cases	Between 10 and 20 cases
	Between 10 and 20 cases Between 50 and 100 cases Other In universities 9% 5 Family 15% 11 In hospitals 9% 5 Constitutional 2% In schools 7% 4 2 Through 7% 4	Between 20 and 50 cases Between 50 and 100 cases Other	Between 20 and 50 cases Between 50 and 100 cases Other
	Immigration private consultancies Refugee and asylum 16% 12 11% 8 HRL 6% 4 Prefer not to 3% 2 say Others 1% 1	In court NGOs Out of court
	% Administrative Total Business and comm. 3% Count 4% 100% 54	3 2	In detention centres In universities Frequency In hospitals
	Criminal			11% 8	Contracts and Oblig. Inheritance In schools
	Always Prefer not to say European Other	33% 2 33% 2	7% 1%	5 1	Property Through private consultancies Family Constitutional Prefer not to say Immigration Refugee and asylum HRL Others
	Once Labour	17% 1	3%	2	Administrative Criminal	Business and comm. European
	Health Translator 17% 1 Intellectual and pat.	4.% 3%	3 2	Other Always Labour Prefer not to say	Banking, bankruptcy Once Translator Environmental
	Total Medical and bio. 100% 6	3%	2	Financial Intellectual and pat. Medical and bio. Health
	Private international 1%	1	Private international Sport law
	Total			100% 74

CULTURAL EXPERTISE IN EUROPE: WHAT IS IT USEFUL FOR? (EURO-EXPERT)

PI: Livia Holden | Data Collector: Ibtisam Sadegh Date of Publication: 01/02/2022

Fields of law

Malta Data Summary CULTURAL EXPERTISE IN EUROPE: WHAT IS IT USEFUL FOR? (EURO-EXPERT)

  PI: Livia Holden | Data Collector: Ibtisam Sadegh Date of Publication: 01/02/2022

	Organisations interested in a teaching program	%	#	Organisations interested in a teaching program
	Know educational	36% 8
	institutions interested
	in teaching
	Interested in teaching	27% 6
	themselves
	Prefer not to say	27% 6
	Know professional	5%	1
	organisations
	interested in teaching
	Other	5%	1
	Total	100% 22	Know educational institutions interested in teaching
				Interested in teaching themselves
				Prefer not to say
				Definitely useful Know professional organisations interested in teaching
				Other	Probably useful
					Might or might not be useful
					Prefer not to say
					22

Reputation of experts

The majority of experts said that they had been regularly instructed/ appointed as expert for many years (60%, 6), followed by 'other' (20%, 2) who indicated to have been recommended by people, and to be fully qualified as court/conference interpreter, then 'the cases in which expert opinion was provided were successful' (10%, 1), and 'I don't know' (10%, 1).

Competitiveness