Stability bounds of a delay visco-elastic rheological model with substrate friction
SStability bounds of a delay visco-elasticrheological model with substrate friction
Malik A. Dawi Jose J. Mu˜noz , ∗ Laboratori de C`alcul Num`eric (LaC`aN)Universitat Polit`ecnica de Catalunya, Barcelona, Spain Dept. of Mathematics, Universitat Polit`ecnica de Catalunya, Barcelona, SpainCentre Internacional de M`etodes Num`erics en Enginyeria (CIMNE), Barcelona, Spain. [email protected]
January 20, 2021
Abstract
Cells and tissues exhibit oscillatory deformations during remodelling,migration or embryogenesis. Although it has been shown that these os-cillations correlate with cell biochemical signalling, it is yet unclear therole of these oscillations in triggering drastic cell reorganisation eventsor instabilities, and the coupling of this oscillatory response with testedvisco-elastic properties.We here present a rheological model that incorporates elastic, viscousand frictional components, and that is able to generate oscillatory re-sponse through a delay adaptive process of the rest-length. We analyseits stability properties as a function of the model parameters and deduceanalytical bounds of the stable domain. While increasing values of thedelay and remodelling rate render the model unstable, we also show thatincreasing friction with the substrate destabilise the oscillatory response.Furthermore, we numerically verify that the extension of the model tonon-linear strain measures is able to generate sustained oscillations thatalternate between stable and unstable regions. keywords:
Oscillations, Delay differential equations, Visco-elasticity,friction , stability, rheology, cells.
Oscillatory cell deformations are ubiquitous and have been quantified in vitro [18, 20] and in vivo , for instance in the segmented clock of mice [27] or during
Drosophila fly dorsal closure [23]. These oscillations have been associated tobiochemical dynamics [13], signalling delays [19] or myosin concentration fluc-tuations [7]. We here present a rheological model that explicitly incorporatesthe delay between the cell length adaptation and the current stretch.1 a r X i v : . [ q - b i o . T O ] J a n ime delay has been included in numerous models in biology, with applica-tions in biochemical negative feedback [15], cell growth and division [1, 11], orcell maturation [10], but are less common in biomechanics. In our case we intro-duce this delay in an evolution law of the cell or tissue rest-length. Such modelswith varying rest-length have been applied to stress relaxation [14], morpho-genesis [5], cortical mechanics [8], or endocytosis [4]. They have the advantageof including a measurable quantity, the rest-length [26], and also furnishing theobserved visco-elastic response. We will here adapt these models and includethe delay response in conjunction with frictional or adhesive forces from theenvironment or substrate.Our visco-elastic model mimics the standard linear solid, but expressed interms of delay rest-length changes, which provides the oscillatory character ofthe deformation. The stability of such system has been described in [17] or in[3] for planar frictionless dynamics of monolayers. We here extend such analysisto a frictional substrate, and deduce the stability conditions as a function ofviscous, stiffness and friction parameters.The stability analysis is usually carried out through the inspection of thecharacteristic equation [2, 22], or semi-discretisation methods [12, 25]. We resortto the former method, and by analysing the associated Lambert function [21,6], we deduce strict and simple bounds of the stability region. We compareour analysis with some numerical solutions of the Delay Differential Equations(DDEs).The article is organised as follows. We describe the visco-elastic model inSection 2 together with the delay evolution law of the rest-length. In Section 3the stability of a linear model is analysed, and some bounds as a function of themodel parameters are given. A non-linear extension is presented in Section 4,which is solved numerically and is analysed with the help of the results obtainedin the linearised model. Our findings are finally discussed in the Conclusionssection. We consider a material rheology that mimics the solid standard mode: a purelyelastic stress σ e in parallel with a visco-elastic stress σ v . Figure 1 shows schemat-ically the two branches. We assume a one-dimensional domain D = [0 , l ( t )], with l ( t ) a time dependent apparent (measurable) length of the domain.The total stress σ in D is given by the sum of elastic and viscoelastic con-tributions, σ = σ e + σ v , where each stress component is given by σ e = k ε ( l ( t ) , l ) and σ v = k ε e ( l ( t ) , L ( t )),with k and k the associated stiffness parameters. The strain measures ε ( l ( t ) , l )and ε e ( l ( t ) , L ( t )) will be detailed in the next sections for the linear and non-linear models. As yet we mention that they depend, in addition to l ( t ), on theinitial length l = l (0) and the rest-length L ( t ) of the visco-elastic branch. This2igure 1: Schematic view of 1-dimensional model, illustrating both elastic andvisco-elastic branches with dissipative friction.rest-length can be interpreted as an internal variable, whose evolution mimicsthe viscous response of Maxwell models [16].More specifically, L ( t ) changes according to the following evolution law˙ L ( t ) = γ ( l ( t − τ ) − L ( t − τ )) , t > . (1)Henceforth we denote by a superimposed dot the time derivatives, i.e. ˙( • ) = d ( • ) /dt . Parameter γ > remodelling rate , which measures the rate atwhich the cell adapts its length to the difference l ( t − τ ) − L ( t − τ ). We haveintroduced the delay parameter τ ≥ σ η with the external substrateor environment, and given by an external force σ η ( t ) = − η ˙ l ( t ), with η ≥ σ η = σ e + σ v readsin our case − η ˙ l ( t ) = k ε e ( l ( t ) , l ) + k ε e ( l ( t ) , L ( t )) , t > , (2)which should be solved together with the evolution law in (1). Due to thepresence of the delay τ , initial conditions must be specified for t ∈ [ − τ, l ( t ) = l , t ∈ [ − τ, , (3) L ( t ) = L , t ∈ [ − τ, , (4)with l and L given constants. In the next sections we will analyse the stabilityand oscillatory regime of the system of Delay Differential Equations (DDE) forlinear and non-linear definitions of the strain measures ε and ε e .3 Stability analysis of linear model
In order to ease the stability analysis, we assume here linear definitions of thestrain measures: ε ( l ( t ) , l ) = l ( t ) − l ,ε e ( l ( t ) , L ( t )) = l ( t ) − L ( t ) . Inserting these expression into the balance equation (2), the set of DDE turninto the following form: − η ˙ l ( t ) = k ( l ( t ) − l ) + k ( l ( t ) − L ( t )) , t > L ( t ) = γ ( l ( t − τ ) − L ( t − τ )) , t > L ( t ) + A L ( t ) + B L ( t − τ ) + c = , t > , (7)with L ( t ) = (cid:26) l ( t ) L ( t ) (cid:27) ; A = (cid:20) k + k η − k η (cid:21) ; B = (cid:20) − γ γ (cid:21) ; c = (cid:26) k l η (cid:27) . Generally, the solution of the coupled system of DDE in (7) is characterizedqualitatively (e.g. asymptotic, synchronous, oscillatory) by the exponents or theroots of the characteristic function [9, 22]. In order to obtain this characteristicfunction, one might search for a solution in the form, L ( t ) = (cid:88) i e m i t L i + L , (8)where L and L i are constant vectors that depend on the chosen initial values,and m i ∈ C are the characteristic exponents. Clearly if all the exponent havenegative real parts, i.e. Re ( m i ) <
0, the solution is asymptotically stable withtime. Substituting Eq. (8) into Eq. (7) gives for each term in the summation (cid:0) m i I + A + B e − m i τ (cid:1) L i = . We remark that the above linear transformation must hold regardless ofthe initial conditions, that is to say, the determinant must always vanish. Thisallows us to express the characteristic function of the system as the determinantof the above matrix, which gives f ( m ) := m + γme − mτ + k + k η m + γk η e − mτ = 0 . (9)4e decompose the characteristic function to real and imaginary parts bysubstituting m = α + iβ and then separating each part, leading to the followingnon-linear system of equations,Re f ( m ) = α − β + k + k η α + γe − ατ (cid:18)(cid:18) α + k η (cid:19) cos( βτ ) + β sin( βτ ) (cid:19) , Im f ( m ) = 2 αβ + k + k η β + γe − ατ (cid:18) β cos( βτ ) − (cid:18) α + k η (cid:19) sin( βτ ) (cid:19) . (10)The stability regions in the parameters space are defined by the borderswhere the number of unstable exponents changes, which means, at least onecharacteristic exponents crosses the imaginary axes from left to right. In suchcase Eq. (10) will have at least one solution with positive α .Here, we have constructed the phase diagram by solving the system in Eq.(10) numerically while monitoring the values of α (see Fig. 2). If there is atleast one root with a positive α the solution was considered unstable.With the aim of furnishing a practical bound for detecting stable solutions,we also give the following result: Proposition 1.
The solution of the system of delay differential equations inEq. (7) with initial conditions in Eq. (3) is stable as long as, k + k − γη − k γτ > . (11) Proof . Condition (11) is derived resorting to the results in [24], and analysingthe so-called
D-curves defined as, R ( ω ) := Re f ( iω ) = − ω + γ (cid:18) k η cos( ωτ ) + ω sin( ωτ ) (cid:19) (12) S ( ω ) := Im f ( iω ) = k + k η ω + γ (cid:18) ω cos( ωτ ) − k η sin( ωτ ) (cid:19) (13)with ω ∈ [0 , + ∞ ). The functions R ( ω ) and S ( ω ) provide infinite parametriccurves that mark the region with constant number of unstable characteristicexponents. In particular, we resort to Theorem 2.19 in [24], which indicatesthat the zeros of Eq. (9) have no real positive parts if and only if, S ( ρ k ) (cid:54) = 0 k = 1 , .., r, (14)and r (cid:88) k =1 ( − k S ( ρ k ) = − , (15)where ρ ≥ ... ≥ ρ r ≥ R ( ω ), with r being anodd number. Moreover, we introduce a polynomial S − ( ω ) which defines a lowerbound for the function S ( ω ) such that,0 < S − ( ω ) ≤ S ( ω ) for ω ∈ (0 , + ∞ ) . (16)5 a) (b) Unstable Stable =0.4=6 =0.4=6=0.5=6 =0.6=6 (c) (d)
Figure 2: Phase diagrams for different pairs of material parameters. (a) Plane( k , k ), (b) plane ( k , η ), (c) plane ( k , η ) and (d) plane ( τ, γ ). The curvesshow stability borders for different values of the off-plane parameters.Continuous lines are obtained with the numerical solution of Eq. (10). Dashedlines represent the sufficient stability condition in Eq. (11). The regions whichare labeled as stable are those with negative values for α and those label as unstable indicate the regions with at least a single positive α .In case that S ( ω ) satisfies the stability conditions in Eq. (14) and (15), S − ( ω ) will also satisfy them by construction. An adequate choice for the poly-nomial S − ( ω ) can be obtained by exploiting the following inequalities,cos( ωτ ) ≥ − , − sin( ωτ ) ≥ − ωτ for ω ∈ (0 , + ∞ )which lead to, S − ( ω ) = (cid:16) k + k η − γ − k η γτ (cid:17) ω. Since ω >
0, the condition in Eq. (16) is satisfied as long as k + k − γη − k γτ > . We point out that the main benefit of Proposition 1 is that it counts in thewhole space of system parameters, giving the opportunity to cross check the6tability taking into account the relative variations of system parameters. Inthe phase diagrams in the parametric space, condition (11) is indicated by thedashed lines in Fig. 2. As it can be observed, it indicates stability regions thatare smaller then those obtained by solving numerically Eq. (10). These plotsemphasise the fact that although the bound in Eq. (16) does not provide anecessary condition, it provides a useful sufficient stability condition.We remark also two salient conclusion from the expression in the bound,which are also confirmed in the phase diagrams: increasing values of γτ have anunstable effect in the lengths l ( t ) and L ( t ), as previously encountered in othermodels [17], while decreasing values of η may render the oscillations stable. Thisis an unexpected result, since increasing viscosity has in general a stabilisingor damping effects in mechanics. This can be explained by highlighting theretardation or delay that viscosity entails in the stress response, similar to anincrease of τ . In order to verify the obtained stability limits, we have preformed some nu-merical tests considering the one-dimensional model presented in Fig. 1. Thetest mimics a previous compression state that is given by the following initialconditions, l ( t ) = L ( t ) = 1 ,τ < t ≤ l ( − τ ) = 0 . , L ( − τ ) = 1 . (18)In order to compare our results with previous values in the literature andwith more general boundary conditions, we will also test different prescribedvalues of l ( t ) and additional external forces. Indeed, in the presence of a constantexternal force f , the equilibrium equation in (2) reads, − η ˙ l ( t ) + f = k ( l ( t ) − l ) + k ( l ( t ) − L ( t )) (19)˙ L ( t ) = γ ( l ( t − τ ) − L ( t − τ )) (20) A backward Euler implicit time discretisation of equations in (19) yields thefollowing set of equations, which are computed sequentially, L n +1 = ∆ tγ ( l n − τ − L n − τ ) l n +1 = 1( η/ ∆ t + k + k ) (cid:16) η ∆ t l n + f n +1 + k L + k L n +1 (cid:17) (21)We here consider the case f n = 0 , n = 0 , , . . . , / ∆ t and ∆ t = 0 . l n and L n is consistent with the stability analysisof the previous section. The presence of the delay τ > l and L , as it can be seen in Fig. 3. The stability of these oscillationsdepends on the model parameters as indicated in the stability diagrams in Fig.2. The first case in Fig. 3a corresponds to stable oscillations, with parametersinside the stability domain, while the second case in Fig, 3b yields unstableoscillations, with parameters that exceed the stability limits. (a) Model parameters: k = 2, k = 3, η = 8, γ = 0 . τ = 6 (b) Model parameters: k = 3, k = 2, η = 8, γ = 0 . τ = 6 Figure 3: Time evolution of current length and rest-length for free unloadedconditions. (a) Parameters belonging to the stable domain. (b) Choice ofparameters that lie outside of the stable domain.
We here choose a constant value of the apparent length l ( t ), with an initialdiscontinuity: L ( t ) = L = 1 , − τ ≤ t ≤ ,l ( − τ ) = 0 . , l = l = 1 , − τ < t. In this case, ˙ l ( t ) = 0 , t >
0, so the the first differential gives us a reactionforce term equal to k ( l − L ( t )), while the DDE reads˙ L = γ ( l − L ( t − τ )) . This DDE (or equivalent forms) has been extensively studied [22, 17], and isknown to yield oscillatory values of rest-length L ( t ) whenever γτ > e , and un-stable oscillations whenever γτ > π . This has been confirmed by the numericalsimulations in Fig. 4. We now impose and external force f = 0 .
2. Since this value only affects thevalue of the vector c in Eq. (5), the stability is consequently unaffected by thevalue of f . The plots in Fig. 5 confirm this fact. These plots show the apparentlength as a function of time, while the rest-length is shown as the contourploton the varying domain x ∈ [0 , l ( t )]. 8 a) Model parameters: γ = 0 . τ = 4 (b) Model parameters: γ = 0 . τ = 5 Figure 4: The evolution of the rest-length with fixed values for the apparentlength l ( t ). The stability is in this case identical to the friction-less models[17]: (a) Oscillatory solution when τ γ > e , (b) unstable solution arisewhenever τ γ > π . (a) Model parameters: k = 1, k = 1, η = 1, γ = 0 . τ = 6 (b) Model parameters: k = 1, k = 1, η = 3, γ = 0 . τ = 6 Figure 5: The evolution of the current length and the rest-length (color map)with prescribed compression forces f ( f ( x = 0) = 0 . , f ( x = 1) = − . We now use a non-dimensional definition of the strains ε ( l ( t ) , l ) = l ( t ) − l l ,ε e ( l ( t ) , L ( t )) = l ( t ) − L ( t ) L ( t ) . While this is a more common strain measure, with non-dimensional values,9hese expressions, when inserted into the equilibrium equations in (2) yield aset of non-linear DDE: − η ˙ l ( t ) = k (cid:18) l ( t ) − l l (cid:19) + k (cid:18) l ( t ) − L ( t ) L ( t ) (cid:19) , (22)˙ L ( t ) = γ ( l ( t − τ ) − L ( t − τ )) . (23)We aim at studying the oscillatory character and stability of these equations.However, due to their non-linearity we cannot directly apply the methodologypreviously presented. We aim instead at analysing the linearised form of equa-tion (22) at time t . By setting δl ( t ) = l ( t ) − l ( t ) and δL ( t ) = L ( t ) − L ( t ),the linear terms read, − ηδ ˙ l ( t ) = k l δl + k L ( t ) δl ( t ) − k l ( t ) L ( t ) δL ( t ) . (24)It then follows that by defining the modified stiffness parameters,ˆ k = k l ( t ) + k L ( t ) (cid:18) − l ( t ) L ( t ) (cid:19) , (25)ˆ k = k l ( t ) L ( t ) , (26)equation (24) is equivalent to the linear terms in the equilibrium equation in(5), but replacing ( k , k ) by (ˆ k , ˆ k ) and in terms of δl ( t ) and δL ( t ) insteadof l ( t ) and L ( t ). This allows us to understand some of the numerical solutionsobtained for the non-linear case.Figure 6a shows the time evolution of l ( t ) and L ( t ), which are sustained, thatis, their asymptotic behaviour does not increase nor decrease. We plot in theparametric space of k and k the modified parameters ˆ k and ˆ k for each time t , as shown in Fig. 6b. It can be observed that although the initial values arelocated in the unstable region, they in turn oscillate between the unstable andstable region, reaching a limit cycle that alternates between the two domains.We have also tested other parameter settings, with an initial location of(ˆ k , ˆ k ) in the parametric space farther from the stability boundary (see Fig.7). In this case, the system exhibits oscillations that reach the singular value L ( t ) = 0 for some t >
0, which renders the DDEs in (22) ill-posed. Instead,when using values that are farther inside the stability region, as it is the case inFig. 8, the oscillations stabilise before reaching this singular value. Althoughwe are not able to furnish bounds for non-linear stability, we can explain thepresence of stable, sustained, or unstable (or singular) oscillations according tothe distance of the initial value of (ˆ k , ˆ k ) to the stability boundary of the linearcase. 10 a) Time evolution of current length andrest-length (b) The evolution of ˜ k and ˜ k . Figure 6: Numerical solution with sustained oscillations of the non-linearmodel. Parameters: k = 1, k = 1, η = 3, γ = 0 . τ = 5 (a) Time evolution of current length andrest-length (b) The evolution of ˜ k and ˜ k Figure 7: Numerical solution with unstable oscillations on the non-linearmodel. Parameters: k = 2, k = 1, η = 3, γ = 0 . τ = 511 a) Time evolution of current length andrest-length (b) The evolution of ˜ k and ˜ k Figure 8: Numerical solution with stable oscillations of the non-linear model.Parameter k = 1, k = 2, η = 3, γ = 0 . τ = 5 Motivated by the presence of delays and visco-elastic response of tissues, we havepresented a rheological model that includes elastic and viscous contributions,and also exhibits oscillatory behaviour.We have analysed the stability of he model when using a linear strain measureand as a function of the model parameters. We have recovered previous results,which show that increasing values of the delay τ and the remodelling rate γ (aquantity that is inversely proportional to tissue viscosity), render the oscillationsunstable. Remarkably, increasing values of the viscous friction of the domainwith respect to external boundary also destabilise the system.By studying the characteristic function of the DDE we have provided suf-ficient conditions of stability and bounds to the stability region. This analysishave also allowed us to explain the presence of sustained oscillations in a non-linear version of the model. This persistent oscillations in the tissue deforma-tions are frequently observed [18, 20], and in our model are due to the transitionbetween stable and unstable domains.We note that despite visco-elastic models based on rest-length changes areincreasingly common [4, 5, 14], their stability in the presence of delayed re-sponse has not been studied. We here provide such an analysis which may alsohelp to explain the observed sudden deformations in embryo development andmorphogenesis. acknowledgements JJM and MD have been financially supported by the Spanish Ministry of Sci-ence, Innovation and Universities (MICINN) with grant DPI2016-74929-R and12y the local government
Generalitat de Catalunya with grant 2017 SGR 1278.
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