Instability and fingering of interfaces in growing tissue
IInstability and fingering of interfaces in growingtissue
Tobias B¨uscher , Angel L. Diez , , Gerhard Gompper and JensElgeti , ∗ Theoretical Soft Matter and Biophysics, Institute of Complex Systems and Institutefor Advanced Simulation, Forschungszentrum J¨ulich, 52425 J¨ulich, Germany Departamento de Estructura de la Materia, Fisica Termica y Electronica, Facultadde Ciencias Fisicas, Universidad Complutense de Madrid, 28040 Madrid, SpainE-mail: [email protected]
Abstract.
Interfaces in tissues are ubiquitous, both between tissue and environmentas well as between populations of different cell types. The propagation of an interfacecan be driven mechanically. Computer simulations of growing tissues are employedto study the stability of the interface between two tissues on a substrate. From amechanical perspective, the dynamics and stability of this system is controlled mainlyby four parameters of the respective tissues: (i) the homeostatic stress (ii) cell motility(iii) tissue viscosity and (iv) substrate friction. For propagation driven by a differencein homeostatic stress, the interface is stable for tissue-specific substrate friction evenfor very large differences of homeostatic stress; however, it becomes unstable above acritical stress difference when the tissue with the larger homeostatic stress has a higherviscosity. A small difference in directed bulk motility between the two tissues sufficesto result in propagation with a stable interface, even for otherwise identical tissues.Larger differences in motility force, however, result in a finite-wavelength instabilityof the interface. Interestingly, the instability is apparently bound by nonlinear effectsand the amplitude of the interface undulations only grows to a finite value in time. a r X i v : . [ q - b i o . T O ] M a r nstability and fingering of interfaces in growing tissue
1. Introduction
Interfaces of tissues, their propagation as well as their stability, play an important role invarious biological contexts, ranging from tissue development [1] to wound healing [2, 3]and cancer [4]. In many of these processes, the interface propagates, driven by cellproliferation and/or motility. This leads to the question how the tissue maintains astable interface, as this is crucial e.g. in development in order to arrive at the desireddistinct cell populations, while interface instabilities can have severe consequences, as incancer metastasis. Several mechanisms act simultaneously in this problem, where eachof them can either have a stabilizing or destabilizing effect on the interface. Interfacialtension, e.g. caused by differential adhesion between cell populations [5], stabilizesan interface, as it penalyzes increase of interface area. On the other hand, increaseof interfacial area can be further amplified, e.g. due to enhanced growth rates in theprotruding region, where cells have more free space and access to nutrients, as commonlyobserved during wound healing [2, 6, 7] .Interface instabilities in systems far from equilibrium are well known in solid-state physics, where several instability mechanisms have been found and studied [8].Examples are the Saffman-Taylor instability (also known as viscous fingering), whichoccurs during the injection of a low-viscosity fluid into one of a larger viscosity, theMullins-Sekerka instability in unidirectional solidification, which arises from the unstablediffusive transport of the latent heat of solidification, and leads to dendritic growth atlater stages, and the Rayleigh-Taylor instability between two immiscible fluids when thefluid with higher density is placed on top of the lighter one. Also, in vapor deposition flatinterfaces are unstable to roughening, in which the interface width initially grows slowly,but monotonically with time and saturates at a finite value at late times. For tissues,or bacterial colonies as a related example, growth and division of cells can give rise tonew instability mechanisms, which, however, may arise from similar mechanisms as the”classical” instabilities of solid-state physics. For example, an undulation instability ofan incompressible epithelium adjacent to a viscoelastic stroma has been found, where theinstability is driven by enhanced growth in the protruding region, which creates a shearflow that builds up pressure at the bottom of the protrusion [9]. Coupling cell growthto nutrient diffusion leads to an additional instability, as cells in the protruding regionhave access to more nutrients, reminiscent of the Mullins-Sekerka instability [10]. Ingrowing bacterial colonies of
E. coli inside a microfluidic device, a streaming instabilityhas been oberseved due to steric interactions between large, slow-moving and small,fast-moving cells [11]. During growth of bacterial colonies on a petri dish, instabilitiesof the advancing front arise, displaying different levels of complexity, which range froma small number of fingers to densely-branched, dentritic structures [12–15].Mechanically regulated propagation of tissues has been studied by employing theconcept of homeostatic stress [16–18]. The homeostatic stress is defined as the stressa tisssue exerts onto its surrounding at the state when apoptosis and division balanceeach other. It has been proposed that in a competition for space between two tissues, nstability and fingering of interfaces in growing tissue
2. Simulation model
Several models have been developed in order to study mechanical properties and growthof cell monolayer in general and interfaces between different cell types in particular[24–26]. For example, vertex-based models are commonly employed , e.g. to studyphysical properties such as shear and compression modulus, or jamming transitions nstability and fingering of interfaces in growing tissue F G ij = G ( r ij + r ) ˆ r ij , (1)with growth-force strength G , unit vector ˆ r ij and distance r ij between the two particlesand a constant r . When the distance between the particles exceeds a threshold r ct the cell divides. A new particle is then placed in close vicinity of each particle of themother cell. These pairs constitute the two daughter cells. Particles between differentcells interact via a soft repulsive force F V ij on short distances and a constant attractiveforce F A ij on intermediate distances, where F V ij = f (cid:16) R r ij − (cid:17) ˆ r ij F A ij = − f ˆ r ij (cid:41) for r ij < R PP , (2)with volume exclusion coefficient f , adhesion strength f and cut-off length R PP . Wemodel apoptosis by removing cells randomly at a constanst rate k a . Interactions withthe underlying substrate are given by a friction force F B i = − γ b v i , (3)with velocity v i . Forces in migrating cell monolayers do not solely arise at the front, butcollectively over the whole monolayer [21, 33]. In a simplifyed picture, this is modeledby a homogeneous bulk motility force [18], given by a constant force perpendicular tothe interface F M i = f m · ˆ e x , (4)with motility-force strength f m and direction ˆ e x perpendicular to the interface. Thischoice of motility model further facilitates comparison of results with Ref. [18]. Adissipative particle dynamics thermostat is employed in order to account for energydissipation and random fluctuation, satisfying the fluctuation-dissipation theorem. Itstemperatute T is chosen low enough that cells can escape local minima, but otherthermal effects are small. Each parameter can be set independently for each cell typeand between cell types for inter-cell interactions.We define a set of standard-tissue parameters and report simulation parameterrelative to these standard values, denoted with a dagger, e.g. G † = G/G (see Tab. S1in the SI for numerical values). Time is measured in terms of the inverse apoptosis rate k a of the standard tissue, distance in terms of the cut-off length R PP and force in unitsof G /R . Thus, the length unit corresponds to the cell size, while time is measured ingenerations. After one time unit, all cells have divided once on average. Quantitiesreported in these units are denoted with an asterisk ∗ . We vary the growth-forcestrength G , the apoptosis rate k a , background friction γ b , and motility-force strength f m . nstability and fingering of interfaces in growing tissue Figure 1.
Simulation snapshots in competitions with different motility-force strengts f A † m of tissue A. The tissues are otherwise identical. The interface moves towardstissue B. From left to right: f A † m = [0 , . , . , . f A † m = 0 . L ∗ y = 80 and time t ∗ = 80 in all. The scale bar is 10 cellsizes. Note that for f A † m ≥ .
002 the snapshots are represantative of the steady stateand the undulations do not grow further.
The cross-adhesion strength between the two tissues is the same as the adhesion strengthwithin one tissue, as reduced cross-adhesion causes enhanced interfacial growth [34].Thus, no passive interfacial tension is present in our simulations.We use the ”treadmilling simulation setup” introduced in Ref. [17] in order to obtainsteady-state interface progression, by keeping the interface position at the center of thesimulation box. All cells are shifted accordingly every 1000 timesteps; excess cells at oneend of the simulation box are removed while the weaker tissue replenishes on the otherend. In this way, both tissues reach their homeostatic state sufficiently far away fromthe interface (with system size L ∗ x = 140 in all simulations), thus the interface propertiescan be studied on long time scales in a computationally efficient way. We measure allquantities in a comoving reference system s = x − x , with interface position x .
3. Results
It was shown in Ref. [17] that the competition between two tissues differing only inhomeostatic stress results in a steady-state interface propagation, where the strongertissue invades the weaker one with a constant velocity. While only stable interfaceswere observed in Ref. [17], Ref. [18] proposes three different routes to instability for aninterface between two competing tissues: (A) For propagation driven by bulk motility,the interface becomes unstable above a critical difference in motility-force strength.For propagation driven by homeostatic stress, the interface is only unstable under thecondition that the two tissues either differ (B) in substrate friction or (C) viscosity. Forboth cases, (B) and (C), the interface becomes unstable above a case-specific criticaldifference in homeostatic stress. For a combination of both, bulk motility force f A/Bm and homeostatic stress difference ∆ σ H = σ BH − σ AH , the interface velocity v int = ∆ σ H + l A ˆ f Am + l B ˆ f Bm ξ A l A + ξ B l B (5) nstability and fingering of interfaces in growing tissue .
005 0 .
010 0 .
015 0 . f A ∗ m − I n t e r f a ce v e l o c i t y v ∗ i n t ∆ σ ∗ H = − . σ ∗ H = − .
16 ∆ σ ∗ H = 0 . σ ∗ H = 0 . − Time t ∗ . . . . . . I n t e r f a ce w i d t h w ∗ ∼ t . ∼ t . f A ∗ m = 0 . f A ∗ m = 0 . f A ∗ m = 0 . f A ∗ m = 0 . (c) (d) .
04 0 . − − Motility force f A ∗ m S a t u r a t i o n w i d t h w ∗ s a t q ∗ peak = 2 · π/L ∗ y q ∗ peak = 3 · π/L ∗ y q ∗ peak = 4 · π/L ∗ y q ∗ peak = 5 · π/L ∗ y − −
10 0 10 20Position s ∗ − . . . O r d e r p a r a m e t e r Q xx f A ∗ m = 0 . f A ∗ m = 0 . f A ∗ m = 0 . f A ∗ m = 0 . Figure 2.
Interface velocity, interface (saturation) width, and order parameterdependence on motility-force strength of tissue A in competitions with with a non-motile tissue. System size L ∗ y = 80 in all, ∆ σ ∗ H = 0 in (b)-(d). (a) Interface velocity v int as a function of the motility-force strength f A ∗ m of tissue A for various homeostaticstress differences ∆ σ ∗ H . Dashed lines represent theoretical predictions according toEq. (5), with parameters fixed by independent simulations. Error bars display standarddeviations (hidden behind markers). (b) Interface width w ∗ as a function of time t ∗ fordifferent values of motility-force strength f A † m of tissue A. Note the logarithmic timescale, the interface width for non-vanishing motility is almost constant for 80% of thetime. (c) Saturation width w ∗ sat as a function of motility-force strength f A ∗ m of tissue Afor different peak wave vectors q ∗ peak . Note the logarithmic scale for f A ∗ m < .
01. Errorbars represent standard deviations. (d) Nematic order paramter Q xx as a function ofthe position s ∗ for various motility-force strengths f A ∗ m of tissue A. Peak wave vector q ∗ peak = 3 · π/L ∗ y for all curves. is predicted, with substrate friction ξ = 2 γ b ρ , cell density ρ , motility-force densityˆ f A/Bm = 2 ρf A/Bm , and stress decay length l = (cid:112) χτ /ξ . Here, χ is the elastic modulus, τ thetime scale at which the tissue loses its elastic character due to cell division and apoptosis,and the product χτ is an effective viscosity. The growth rate k is expanded to linearorder around the homeostatic stress as k = κ ( σ − σ H ), with stress-response coefficient κ .The viscosity is connected to the stress-response coefficient via κ = 1 /χτ . For oursimulations, these coarse-grained tissue parameters are either direct input parameters,or can be measured in independent single tissue simulations.Figure 2(a) displays a comparision between Eq. (5), with parameters fixed byindependent simulations (see Refs. [17, 35] for details), and the measured interface nstability and fingering of interfaces in growing tissue We study first the effect of bulk motility without additional difference in homeostaticstress, i.e. the two tissues are identical except that tissue A has a motility force f Am > f Bm = 0). As predicted in Ref. [18], a prescribed motilityforce can drive interface propagation and the motile tissue invades the non-motile oneat constant velocity. An instability is predicted for∆ v f > l A ξ A + l B ξ B ) l A l B ξ A ξ B ( l A + l B ) , (6)with difference in bulk velocity ∆ v f = f Am /ξ A − f Bm /ξ B and interfacial tension Γ [18].Figure 1 displays simulation snapshots for increasing motility-force strength oftissue A. For vanishing motility force, the two competing tissues are identical, includingthe interaction between cells of different tissues, and thus the interface width w ( t ) = (cid:112) (cid:104) h (cid:105) − (cid:104) h (cid:105) (with height field h ( y, t ), see Ref. [17] for more details) diverges as afunction of time (see snapshots in Fig. 1 and blue line in Fig. 2(b), as well as Vid. S1 inthe SI). However, a rather small motility-force strength of tissue A ( f Am ≈ · − G /R )suffices to arrive at a finite interface saturation width w sat , i.e. small motility forces havea stabilizing effect on the interface (see snapshots in Fig. 1 and green line in Fig. 2(b), aswell as Vid. S2 in the SI). For larger motility-force strengths, protrusions of the motileinto the non-motile tissue form at one particular finite wavelength. Over the time courseof the first cell generation the interface width grows slowly with time ( w ∼ t . ). Afterthe unstable wave mode has been selected, the interface width increases linear with time( w ∼ t . ). However, the mode amplitude does not grow indefinitely, but saturates at amotility-force dependent plateau due to nonlinear effects after about ten cell generations(see snapshots in Fig. 1 and orange and red line in Fig. 2(b), as well as Vids. S3 andS4 in the SI).The resulting wave pattern is remarkably stable over time once the steady statehas been reached. Figure 2(c) displays the saturation width w sat as a function of themotility-force strength. The saturation width first decreases with increasing motility-force strength, with w sat of the order of one or two cell layers at the minimum, i.e. analmost flat interface. For higher motility-force strength, the saturation width starts toincrease and the aforementioned protrusions form, which we interpret as the onset ofinstability. Interestingly, independent simulations for identical parameter yield differentwavelengths at the steady state. While the saturation width decreases with increasing q peak for identical f Am , the smallest motility-force strength at which a particluar wavemode is found increases with q peak . This matches the predicted evolution of the mostunstable wave mode in Ref. [18].In order to study the observed interface patterns quantitatively, we calculate the nstability and fingering of interfaces in growing tissue − Wave vector q ∗ S tr u c t u r e f a c t o r S ( q ∗ ) ∼ q − . f A ∗ m = 0 . f A ∗ m = 0 . f A ∗ m = 0 . (b) − Wave vector q ∗ S tr u c t u r e f a c t o r S ( q ∗ ) q ∗ peak = 3 · π/L ∗ y q ∗ peak = 4 · π/L ∗ y q ∗ peak = 5 · π/L ∗ y Figure 3. (a) Structure factor S ( q ∗ ) at the steady state for different values of themotility-force strength f A ∗ m of tissue A for ∆ σ ∗ H = 0 and q ∗ peak = 3 · π/L ∗ . Thedashed line is a guide to the eye. (b) Same as (a), but for fixed motility-force strength f A ∗ m = 0 .
55 and different peak wave vectors q ∗ peak . (c) Snapshots obtained in thesimulations of b) at the steady state at t ∗ = 80. Note that the different stable peakwave vectors arise by chance from an initially flat interface. System size L ∗ y = 80 inall. time-averaged structure factor S ( q ) = (cid:104) ˜ h ( q, t )˜ h ( − q, t ) (cid:105) , (7)at the steady state, where ˜ h ( q, t ) denotes the spatial Fourier transform of the heightfield h ( y, t ) (see Ref. [17] for further details). For self-affine surface growth the structurefactor displays a power-law decay at the steady state [26, 36]. Figure 3(a) shows thestructure factor for the same values of motility-force strength as in Figs. 1 and 2(b). S ( q ) displays deviations from a power-law decay by a peak at a certain wave vectorlarger than the system-spanning one (in Fig. 3(a) q peak = 3 · π/L ), which gets morepronounced for increasing motility-force strength and corresponds to the wavelength ofthe protrusions in Fig. 1. As mentioned above, for the same value of f Am , different wavevectors can become the dominating mode at the steady state in independent simulations. nstability and fingering of interfaces in growing tissue q peak (see Fig. 2(c)).The stabilizing effect is accompanied by a preferred alignment of cells perpendicularto the interface, quantified by the nematic order paramter Q xx = 2 p x p x −
1, with p x the x -component of the unit vector between the two cell particles. This leadsto an active interfacial tension Γ = (cid:82) ∞−∞ ( σ yy ( s ) − σ xx ( s ))d s , due to cell growth [17].Figure 2(d) displays the order paramter for different motility-force strengths. The overallalignment along the y -direction (i.e. negative Q xx ) first increases with growing f Am , witha maximum at the interface position. In the regime where protrusions start to form,the maximum splits into two maxima located to the left and the right of the interface,where the position of the maxima corresponds to the width of the protrusions. Foreven higher motility-force strength, when the saturation width becomes large, we findan overall alignment along the x -direction. As shown in Refs. [17, 18], interface propagation can be driven by homeostatic stressalone. For two tissues that only differ by their homeostatic stress, a stable interfacepropagating at constant velocity is found in the simulations [17]. Two instabilityconditions for competition driven by a difference in homeostatic stress ∆ σ H have beenproposed in Ref. [18], given by∆ σ H >
274 Γ ( ξ A l B − ξ B l A ) ( ξ A l A + ξ B l B ) l l ( ξ B − ξ A ) , ξ B > ξ A (8)∆ σ H >
2Γ ( ξ A l A + ξ B l B ) κ − − κ − , κ − > κ − (9)While substrate friction ξ can be changed as an input parameter, the stress-responsecoefficient κ is a tissue property, which needs to be determined in simulations and cannot be controlled directly. In order to measure κ , we use a constant-stress ensembleand measure the growth rate as a function of the applied stress. κ is then obtained bya linear fit [17, 35]. Since κ = 1 /χτ , with the characteristic time τ for cell turnover, κ can be changed by varying the apoptosis rate k a . Reduction of k a yields a lowerstress-response coefficient and thus a higher viscosity.For different substrate frictions, we do not observe any instabilities, even for largedifferences in homeostatic stress. While the overall saturation width increases withgrowing homeostatic stress difference, w sat does not show systematic variations withsubstrate friction (see Fig. 5(a)).According to Eq. (9), instabilities should only be obtained if the weaker tissue (thetissue with the higher homeostatic stress, here tissue B) has a larger viscosity, i.e. alower apoptosis rate than the stronger tissue. Figure 4 displays simulation snapshotsfor different homeostatic stress differences. With increasing difference, a finger of the nstability and fingering of interfaces in growing tissue Figure 4.
Simulation snapshots in competitions with different homeostatic stressdifferences ∆ σ H and reduced apoptosis rate k B † a = 0 . σ H = [0 . , . , . , . L ∗ y = 80 and time t ∗ = 80 in all. The scale bar is 10 cell sizes. The snapshots arenot represantative of a steady state, as fingers detach, disappear and reform over time. (a) (b) γ B † b S a t u r a t i o n w i d t h w ∗ s a t ∆ σ ∗ H = 0 . σ ∗ H = 0 . σ ∗ H = 0 . σ ∗ H = 0 . σ ∗ H = 1 . k B † a S a t u r a t i o n w i d t h w ∗ s a t ∆ σ ∗ H = 0 . σ ∗ H = 0 . σ ∗ H = 0 . σ ∗ H = 0 . σ ∗ H = 1 . Figure 5. (a) Saturation width w ∗ sat as a function of substrate friction γ A † b of tissue Afor various homeostatic stress differences ∆ σ ∗ H . (b) Same as (a) but as a function ofthe apoptosis rate k B † a of tissue B. System size L ∗ y = 80 in both. Error bars representstandard deviations. Note the different scales on the y -axis between (a) and (b). weaker tissue is found to develop into the stronger one. In contrast to the motility-driven case, no steady state is reached. The finger occasionally detaches, leaving alarge island behind in the stronger tissue, moves along the interface and forms again(see Vid. S5 in the SI). However, we still find a mostly stable saturation width of theinterface. Figure 5(b) displays w sat as a function of the apoptosis rate k Ba of tissue B forvarious different values of ∆ σ H . We find that w sat increases for a reduced apoptosis rate(compared to k Ba = k Aa ) above a critical homeostatic stress difference (∆ σ ∗ H ≈ . − . k Ba , i.e. an enhanced apoptosis rateof the weaker tissue has a stabilizing effect.The structure factor reflects the increase in saturation width with growinghomeostatic stress difference (see Fig. 6). Below the critical stress difference, thestructure factor for reduced apoptosis rate does not deviate significantly from the caseof identical apoptosis rates of the competing tissues (see Fig. 6(a)). However, for a fixed(reduced) apoptosis rate of tissue B, the amplitude of all wave modes increases withgrowing ∆ σ H (see Fig. 6(b)), which matches the increase of the interface saturationwidth. nstability and fingering of interfaces in growing tissue − Wave vector q ∗ S tr u c t u r e f a c t o r S ( q ∗ ) ∼ q − . ∼ q − . k B † a = 0 . k B † a = 0 . k B † a = 1 . − Wave vector q ∗ S tr u c t u r e f a c t o r S ( q ∗ ) ∼ q − . ∼ q − . ∆ σ ∗ H = 0 . σ ∗ H = 0 . σ ∗ H = 0 . Figure 6. (a) Structure factor S ( q ∗ ) at the steady state for different values of theapoptosis rate k B † a of tissue B for ∆ σ ∗ H = 0 .
18 (below the critical stress difference).The dashed lines are guides to the eye. (b) Same as (a), but for different values of∆ σ ∗ H and fixed k B † a = 0 .
2. System size L ∗ y = 80 in all. Finally, we take a closer look at a combination of differences in motility and homeostaticstress, with substrate friction and apoptosis rate identical for both tissues. For smallmotility forces, the results of Ref. [17] are not altered, the interface is stable andpropagates at a constant velocity. In the regime where we find protrusions of the motiletissue into the non-motile tissue for vanishing homeostatic stress difference, the interfacesaturation width likewise starts to increase (see Fig. 7(a)). However, we do not observeprotrusions at a particular wave length as for ∆ σ ∗ H = 0, but a highly dynamic shape ofthe interface (see snapshots in Fig. 7(b) for a comparision and Vid. S6 in the SI).
4. Discussion
We have investigated the stability of a propagating interface between two competingtissues over a broad parameter range in simulations of a particle-based model.While the width of an interface between two tissues with identical propertiesdiverges as a function of time, we find that already a very small directed bulk motilityforce of one tissue suffices to stabilize the interface at a finite width, similar to ahomeostatic stress difference [17]. Above a critical motility-force strength, a single modewith wave length less than the system size becomes unstable. However, the amplitudeof this mode does not diverge, as expected by linear-stability analysis, but nonlineareffects limit its growth, resulting in remarkably stable steady-state undulations of theinterface. Cells align preferentially parallel to the interface for small motility forces,which transists into perpendicular alignment with growing motility-force strength.For interface propagation driven by a difference in homeostatic stress, an enhancedviscosity due to a reduced apoptosis rate of the weaker tissue results in an unstableinterface above a critical homeostatic stress difference, reminiscent of a Saffman-Taylorinstability. The resulting pattern is much more dynamic than in the motility-driven nstability and fingering of interfaces in growing tissue .
04 0 . − − Motility force f A ∗ m S a t u r a t i o n w i d t h w ∗ s a t ∆ σ ∗ H = − . σ ∗ H = − . σ ∗ H = 0 . Figure 7. (a) Saturation width w ∗ sat as a function of the motility-force strength f A ∗ m of tissue A for various homeostatic stress differences ∆ σ ∗ H in competitions with thestandard tissue with f B ∗ m = 0 and system size L ∗ = 80. Note the logarithmic scale for f A ∗ m < .
01. Error bars represent standard deviations. (b) Simulation snapshots fordifferent motility-force strengts f A ∗ m of tissue A, without (left) and with a homeostaticstress difference ∆ σ ∗ H = 0 .
18 (right). From top to bottom: f A † m = [0 . , . , .
08] Theinterface moves to the right. The tissues are otherwise identical. System size L ∗ y = 80and time t ∗ = 80 in all. Note that the snapshots with a homeostatic stress differenceare not represantative of the steady state, as the interface shape is highly dynamic. case. A finger of the weaker tissue remains within the propagating front. This fingerconstantly reforms, moves and disappears.These two instabilities have recently been predicted by linear-stability analysis [18].For both instabilities, our results match the predicted evolution of the most unstablewave mode qualitatively. However, a quantitative comparision remains elusive, sincewe do not consider interfacial tension due to differential adhesion, as this would causeinterfacial growth due to more free space for cells at the interface [34].However, we do not observe the predicted instability for a difference in substratefriction of the competing tissues, even for large differences in the homeostatic stressbetween the competing tissues.Our results suggest that interfacial patterns of competing tissues provideinformation about the underlying mechanical properties of the competing tissues. Forexample, a relatively regular — almost sinusoidal — undulation pattern would suggesta motility-driven invasion, whereas a ”remaining finger” of the host in the invadingtissue would indicate a lower viscosity of the invader. However, experimental evidenceof this kind of structures and instabilities will be needed before definite conclusionscan be drawn. From a theoretical perspective, possible future research directions onthe stability of interfaces could be to account for anisotropic cell growth or enhancedinterfacial growth rates [34]. nstability and fingering of interfaces in growing tissue
5. Acknowledgements
The authors gratefully acknowledge the computing time granted through JARA-HPC on the supercomputer JURECA [37] at Forschungszentrum J¨ulich. Partsof the simulations were performed with computing resources granted by RWTHAachen University under project rwth0475.
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