INITIAL LAYER ANALYSIS FOR A LINKAGE DENSITY IN CELL ADHESION MECHANISMS

In this paper we present a non local age structured equation involved in cell motility modeling [5,9,11]. It describes the evolution of a density of linkages of a point submitted to adhesion. It depends on an asymptotic parameter ε representing the characteristic age of linkages. Here we introduce a new initial layer term in the asymptotic expansion with respect to ε. This improves error estimates obtained in [5]. Moreover, we study the convergence of the time derivative of this density and show how a singular term appears when ε goes to zero. We show convergence, in the tight topology of measures, to the time derivative of the limit solution and a Dirac mass supported on the initial half-axis. In order to illustrate these results, numerical simulations are performed and compared to the asymptotic expansion for various values of ε. Résumé. Dans cet article, nous analysons un problème structuré en âge avec un terme non local de saturation. Ce problème apparait dans la modélisation de la motilité et des mécanismes d’adhérence cellulaires [5, 9, 11]. L’équation dépend d’un paramètre asymptotique ε représentant l’âge caractéristique des liaisons. Ici, nous introduisons un nouveau terme de couche initiale dans le développement asymptotique par rapport à ε. Ceci améliore les estimations d’erreur obtenues dans [5]. En outre, nous étudions la convergence de la dérivée en temps de la densité des liaisons et montrons comment un terme singulier apparait quand ε devient nul. Nous montrons la convergence, dans la topologie étroite des mesures, vers la somme de la dérivée temporelle de la solution limite et d’une masse de Dirac supportée par le demi-axe initial. Des simulations numériques sont effectuées pour diverses valeurs d’ε et comparées au développement asymptotique afin d’illustrer ces résultats théoriques. Introduction This work is related to the mathematical modeling of cell motility. Originally, a mechanical description of a network of actin filaments allowed to model the lamellopodium and was presented in [9–11]. From these, D. Oelz and the author extracted a toy model presented and extensively analyzed from the mathematical point of view in [5]. The model describes the motion of a single adhesion point submitted to some ascribed external force. This motion is described by zε, a position variable solving a Volterra equation coupled with a density of linkages ρε weighting the adhesions already set and creating the pull-back force exerted from these (see (1) below). Then various mathematical problems were handled extending [5] : in [6], the authors weakened some of the hypotheses and rephrased the model in new variables, while in [7] a non-linear coupling was exposed and blow-up in finite time vs global existence was proved. More recently, a model closer to the original description in [9] is analyzed [4]. This led to new questions, this work aims to answer one of those. We focus, here, on the ∗ Mohammed Boubekeur performed the numerical simulations illustrating the main results of the paper. 1 Laboratoire Analyse & Géométrie, Université Villeteuse Paris 13, France c © EDP Sciences, SMAI 2018 Article published online by EDP Sciences and available at https://www.esaim-proc.org or https://doi.org/10.1051/proc/201862108 ESAIM: PROCEEDINGS AND SURVEYS 109 age structured model governing the density of linkages ρε. The age distribution ρε = ρε(a, t) is the solution of the following system :  ε∂tρε + ∂aρε + ζε ρε = 0 , a > 0 , t > 0 , ρε(0, t) = βε(t) (1− μ0,ε) , a = 0, t > 0 , ρε(a, 0) = ρI(a) , a > 0, t = 0, (1) where μk,ε(t) := ∫∞ 0 aρε(ã, t) dã and the on-rate of bonds is a given coefficient βε times a factor, that takes into account saturation of the moving binding site with linkages. The off-rate ζε is a prescribed function representing the death rate of the population for a given age a at a time t. The limit function ρ0 := limε→0 ρε is explicitly given by ρ0(a, t) = 1 1 β0(t) + ∫∞ 0 exp ( − ∫ b 0 ζ0(ã, t) dã ) db exp ( − ∫ a 0 ζ0(ã, t) dã ) , (2) being the solution of  ∂aρ0 + ζ0 ρ0 = 0 , t ≥ 0 , a > 0 , ρ0(a = 0, t) = β0(t) ( 1− ∫ ∞ 0 ρ0(ã, t) dã ) , t ≥ 0 . (3) The characteristic curves associated to (1) are straight lines parallel to t = εa. Close to S0 := {(a, t) ∈ R+ × {0}}, there is a set Sε := {(a, t) ∈ (R+) ; εa > t}, where the initial condition is transported and when ε goes to zero, Sε collapses to S0. The paper focusses precisely on the behavior of ρε in this area. Namely in [5], the convergence of ρε towards ρ0 was shown in C((0, T ];L(R+)) ∩ L((0, T )× R+) : an a priori estimate was obtained leading to ‖ρε(·, t)− ρ0(·, t)‖L1(R+) ≤ C exp(−ζmint/ε) + oε(1). (4) where ζmin is the strictly positive lower bound of ζε. Here, we enrich the asymptotic expansion with a supplementary term. We solve the initial layer problem : find ρ̃0 solving  ∂t̃ρ̃0 + ∂aρ̃0 + ζ0(a, 0)ρ̃0 = 0, (a, t̃) ∈ (R+), ρ̃0(0, t̃) = −β0(0) ∫ R+ ρ̃0(a, t̃)da, a = 0, t̃ > 0, ρ̃0(a, 0) = ρI(a)− ρ0(a, 0) =: ρ̃I(a), a > 0, t̃ = 0. (5) This improves the error estimates above. Indeed for any t ≥ 0, one has now : ‖ρε(·, t)− ρ0(·, t)− ρ̃0(·, t/ε)‖L1(R+) . oε(1). The initial layer is amicroscopic lifting of the initial condition of the difference ρε−ρ0, it removes the exponential decay occurring in Sε, which is visible on the first term of the right hand side in (4). As the boundary term in (1) is non local, the analysis is not straightforward, and relies strongly on the specific energy functional introduced in [5]. Since ε mutiplies ∂tρε in (1), it is interesting to investigate to which space the time derivative ∂tρε belongs uniformly with respect to ε. Another question of interest is to characterize its limit when ε goes to zero. We show in Theorem 3.2 that ∂tρε belongs to the dual of Cb([0, T ] × R+) (bounded continuous functions on [0, T ] × R+) uniformly with respect to ε. Moreover, it tends to ∂tρ0 and a Dirac mass supported by S0 in the tight topology of measures (see Theorem 3.3 for a precise claim). The paper is organized as follows : in Section 1 we present some notations, the main assumptions and useful results from [5], in Section 2, we analyse the initial layer (5), provide basic existence results. We also show that ρ̃0 is of bounded variation in time, so that the time derivative is related to a bounded Radon measure. We exhibit some limits involving this measure used later on. In Section 3, we construct the zero order asymptotic 110 ESAIM: PROCEEDINGS AND SURVEYS expansion and show error estimates, we then prove that ∂tρε is also associated to a finite Radon measure and finally we provide error estimates in the total variation norm and show the main claim in Proposition 3.2 and Theorem 3.3. Numerical simulations are performed in order to illustrate these results in Section 4. 1. Notations, hypotheses and previous results 1.1. Some notations We set QT := R+ × (0, T ) and QT := R+ × [0, T ]. The space of signed, locally bounded Radon measures M loc(QT ) is by Riesz’ Theorem identified as the space of linear forms on Cc(QT ), the space of continuous functions with compact support in QT . We call C0(QT ) the space of continuous functions vanishing at infinity. The space of signed bounded Radon measures is denoted M(QT ) = C0(QT )∗ (for more details cf [1, 3, 8] and references therein). We define Cb(QT ), the space of bounded continuous functions on QT . The absolute value applied to a measure denotes the total variation measure, i.e. if λ ∈M loc(QT ) then by the Hahn decomposition, λ = λ+ − λ− and |λ| = λ+ + λ− where λ± are positive measures. For any function u defined a.e. (a, t) ∈ QT , we define the discrete time derivative operator D t D t u(a, t) := u(a, t+ τ)− u(a, t) τ (6) where τ is a small positive parameter. In what follows, we put ourselves in a similar context as in [5]. For this sake we recall assumptions and main results useful for the rest of the paper. 1.2. Main assumptions Assumptions 1.1. The dimensionless parameter ε > 0 is assumed to induce two families of chemical rate functions that satisfy: (i) For any T > 0 the function βε is Lipschitz in [0, T ] (the Lipschitz constant is denoted βLip) and ζε is in Lipt([0, T ];L ∞ a (R+)) (resp. ζLip). (ii) For limit functions β0 ∈W 2,∞([0, T ]) and ζ0 ∈W 2,∞([0, T ];La (R+)), moreover it holds that ‖ζε − ζ0‖Lipt([0,T ];L∞(R+)) → 0 and ‖βε − β0‖W 1,∞([0,T ]) → 0 as ε→ 0. (iii) We also assume that there are upper and lower bounds such that 0 < ζmin ≤ ζε(a, t) ≤ ζmax and 0 < βmin ≤ βε(t) ≤ βmax for all ε > 0, a ≥ 0 and t > 0. The initial data for the density model (1) satisfies some hypotheses that we sum up here: Assumptions 1.2. The initial condition ρI ∈ La (R+) satisfies (i) positivity ρI(a) ≥ 0 , a.e. in R+ , moreover, one has also that the total initial population satisfies 0 < ∫ R+ ρI(a)da < 1 . ESAIM: PROCEEDINGS AND SURVEYS 111 (ii) boundedness of higher moments, 0 < ∫ R+ aρI(a) da ≤ cp , for p = 1, 2 , where cp are positive constants depending only on p. (iii) further regularity : we assume that ∂aρI ∈ L(R+), which together with the first hypotheses on ρI implies that ρI ∈W (R+). 1.3. Useful existing results In this setting, one has existence and uniqueness as stated in Theorem 2.1 [5] recalled here for sake of self-compliance. Theorem 1.1. Let assumptions 1.1 and 1.2 hold, then for every fixed ε there exists a unique solution ρε ∈ C(R+;L(R+)) ∩ L((R+)) of the problem (1). We say that ρε is a mild solution since it satisfies (1) in the sense of characteristics, namely ρε(a, t) =  βε(t− εa) ( 1− ∫∞ 0 ρε(ã, t− εa) dã ) × × exp ( − ∫ a 0 ζε(ã, t− ε(a− ã)) dã ) , when a < t/ε , ρI(a− t/ε) exp ( − 1ε ∫ t 0 ζε((t̃− t)/ε+ a, t̃) dt̃ ) , if a ≥ t/ε . (7) Moreover, it is a weak solution as well since it satisfies ∫ ∞


Introduction
This work is related to the mathematical modeling of cell motility. Originally, a mechanical description of a network of actin filaments allowed to model the lamellopodium and was presented in [8][9][10]. From these, D. Oelz and the author extracted a toy model presented and extensively analyzed from the mathematical point of view in [4]. The model describes the motion of a single adhesion point submitted to some ascribed external force. This motion is described by z ε , a position variable solving a Volterra equation coupled with a density of linkages ρ ε weighting the adhesions already set and creating the pull-back force exerted from these (see (2) below). Then various mathematical problems were handled extending [4] : in [5], the authors weakened some of the hypotheses and rephrased the model in new variables, while in [6] the non-linear coupling was exposed and blow-up in finite time vs global existence was proved.
More recently, a model closer to the original description in [8] is analyzed [3]. This led to new questions, this work aims to answer one of those. We focus, here, on the age structured model governing the density of linkages ρ ε . The age distribution ρ ε = ρ ε (a, t) is the solution of the following system : where µ k,ε (t) := ∞ 0 a k ρ ε (ã, t) dã and the on-rate of bonds is a given coefficient β ε times a factor, that takes into account saturation of the moving binding site with linkages. The off-rate ζ ε is a prescribed function representing the death rate of the population for a given age a at a time t. The limit function ρ 0 := lim ε→0 ρ ε is explicitly given by being the solution of The characteristic curves associated to (2) are straight lines parallel to t = εa. Close to S 0 := {(a, t) ∈ R + × {0}}, there is a set S ε := {(a, t) ∈ (R + ) 2 ; εa > t}, where the initial condition is transported and when ε goes to zero, S ε collapses to S 0 . The paper focusses precisely on the behavior of ρ ε in this area. Namely in [4], the convergence of ρ ε towards ρ 0 was shown in C((0, T ]; L 1 (R + )) ∩ L 1 ((0, T ) × R + ) : an a priori estimate was obtained leading to where ζ min is the strictly positive lower bound of ζ ε . Here, we enrich the asymptotic expansion with a supplementary term. We solve the initial layer problem : findρ 0 solving This improves the error estimates above. Indeed for any t ≥ 0, one has now : The initial layer is a sort of microscopic lifting of the initial condition of the difference ρ ε − ρ 0 , it removes the exponential decay occurring in S ε , which is visible on the first term of the rhs in (5). As the boundary term in (2) is non local, the analysis is not straightforward, and relies strongly on the specific energy functional introduced in [4]. Since ε mutiplies ∂ t ρ ε in (2), it is interesting to investigate to which space the time derivative ∂ t ρ ε belongs uniformly wrt ε. Another question of interest is to characterize its limit when ε goes to zero. We show in Theorem 3.2 that ∂ t ρ ε belongs to the dual of C b ([0, T ] × R + ) (bounded continuous functions on [0, T ] × R + ) uniformly wrt ε. Moreover, it tends to ∂ t ρ 0 and a Dirac mass supported by S 0 in the tight topology of measures (see Theorem 3.3 for a precise claim).
The paper is organized as follows : in Section 1 we present some notations, the main assumptions and useful results from [4], in Section 2, we analyse the initial layer, provide basic existence results and show that it is of bounded variation in time, so that the time derivative is related to a bounded Radon measure. We show some limits involving this measure used later on. In Section 3, we construct the zero order asymptotic expansion and show error estimates, we then prove that ∂ t ρ ε is also associated to a finite Radon measure and finally we provide error estimates in the total variation norm and show the main claim in Proposition 3.2 and Theorem 3.3. Numerical simulations are performed in order to illustrate these results in Section 4 1. Notations, hypotheses and previous results

Some notations
We set Q T := R * + × (0, T ) and Q T := R + × [0, T ]. The space of signed, locally bounded Radon measures M 1 loc (Q T ) is by Riesz' Theorem identified as the space of linear forms on C c (Q T ), the space of continuous functions with compact support in Q T . We call C 0 (Q T ) the space of continuous functions vanishing at infinity. The space of signed bounded Radon measures is denoted M 1 (Q T ) = C 0 (Q T ) * (for more details cf [1,7] and references therein). We define C b (Q T ), the space of bounded continuous functions on Q T . The absolute value applied to a measure denotes the total variation measure, i.e. if λ ∈ M 1 loc (Q T ) then by the Hahn decomposition, λ = λ + − λ − and |λ| = λ + + λ − where λ ± are positive measures. For any function u defined a.e. (a, t) ∈ Q T , we define the discrete time derivative operator D τ t D τ t u(a, t) := u(a, t + τ ) − u(a, t) τ where τ is a small positive parameter.
In what follows, we put ourselves in a similar context as in [4]. For this sake we recall assumptions and main results useful for the rest of the paper.

Main assumptions
Assumptions 1.1. The dimensionless parameter ε > 0 is assumed to induce two families of chemical rate functions that satisfy: We also assume that there are upper and lower bounds such that for all ε > 0, a ≥ 0 and t > 0.
The initial data for the density model (2) satisfies some hypotheses that we sum up here: a.e. in R + , moreover, one has also that the total initial population satisfies (ii) boundedness of higher moments, where c p are positive constants depending only on p.
(iii) further regularity : we assume that ∂ a ρ I ∈ L 1 (R + ), which together with the first hypotheses on ρ I implies that ρ I ∈ W 1,1 (R + ).

Useful existing results
In this setting, one has existence and uniqueness as stated in Theorem 2.1 [4] recalled here for sake of self-compliance.  (2). We say that ρ ε is a mild solution since it satisfies (2) in the sense of characteristics, namely Moreover, it is a weak solution as well since it satisfies for every T > 0 and every test function Lemma 1.1. Let assumptions 1.1 and 1.2 hold, then the unique solution ρ ε ∈ C(R + ; Furthermore the following results on the convergence of ρ ε as ε tends to 0 have been obtained. We define the functional and we obtain Lemma 1.2. Let ζ min > 0 be the lower bound to ζ ε (a, t) according to assumption 1.1, then it holds for all t ≥ 0 that As a consequence we conclude Theorem 1.2. Let ρ ε be the solution to the system (2) according to Theorem 1.1 and let the ρ 0 be as defined in (3), then it holds that Remark 1.1. Note that in general ρ ε,I does not converge to ρ 0 (·, 0) in L 1 a as ε → 0. An initial layer will be observable and its profile will be shaped like a multiple of e −ζ min t ε , which is again a consequence of Lemma 1.2.

The initial layer
The limit solution ρ 0 can be seen as a crude approximation of ρ ε for a fixed ε. Since ρ 0 does not depend on ρ I , in a certain way it has forgotten the past. This is why we introduce a microscopic corrector solving (6).

Existence uniqueness and a priori estimates
The same analytic tools as above are applied in order to prove existence and uniqueness ofρ 0 on compact time intervals. We detail the global existence and boundedness.
Proof. Local existence in C([0, T ]; L 1 (R + )) is easy and follows the same lines as in Theorem 1.1. The global existence in L 1 (R + × R + ) is more involved and we detail it here. Using the functional H one has that which by applying Grönwall's Lemma gives that and this integrated in time gives the . Inequality (13) also proves that µ 0 ∈ L ∞ (R + ) and thusρ 0 ∈ L ∞ (R + × R + ) since the initial and boundary values are bounded. Since on any compact interval in timeρ 0 is continuous with values in L 1 a (R + ), the claim follows. Proof. The proof is a consequence of (13), which multiplied by t and integrated in time provides the claim.

2.2.
The time derivative ∂ tρ0 is a signed bounded Radon measure Theorem 2.2. If ζ ε (·, 0) is a function bounded from below, the time derivative of the initial layer ∂ tρ0 can be identified with a finite Radon measure λ ∂tρ0 and one has where the absolute value of the measure denotes its total variation. One has also that The proof is very similar to the proof of Theorem 3.2, and is left to the reader.
Proof. The convergence occurs in the weak topology, i.e. for all φ ∈ C 0 (Q T ) (cf p.11 Chap. 4, [7]). The continuous linear form extends to all φ ∈ C b (Q T ), (see Lemma 3.3.2, p.11 [7]). The convergence occurs also in the tight topology of measures. Indeed, we consider By Lebesgues theorem as R goes to infinity the norm goes to zero. This shows that for every δ > 0 there exists which implies that λ D τ tρ 0 tends towards λ ∂tρ0 in the tight topology up to the extraction of a subsequence (cf Lemma 3.4.5, p.14 [7]).

Moreover one has
Proposition 2.2. Under the same hypotheses as above, one has Proof. Becauseρ 0 is of bounded variation wrt time This shows that q(t) := R+ ϕ(a)ρ 0 (a, t)da is a function of bounded variation, thus there exists a Radon measure λ ∂tq associated to the time derivative of q. The integral T /ε 0 dλ ∂tq coincides with the Riemann-Stieltjes integral. Thus integration by parts holds, leading to On the other hand, D τ t and integration commute giving that By Proposition 2.1, the lhs tightly converges to its limit, whereas the rhs converges weakly, leading to which thus gives that The first term in the rhs can be estimated as which vanishes as ε goes to zero since T > 0.

Error estimates on the density of linkages
Simple computations on the explicit solution ρ 0 of (4) given by (3) for all (a, t) in Q T .
Using Grönwall's Lemma and the fact that H[ρ ε (·, 0)] = 0 one has Now we detail the second term in the rhs, the first follows the same arguments : where we assumed that β ε is uniformly Lipschitz in [0, T ] according to hypotheses 1.1, and we used Theorem 2.1 and its Corollary 2.1.
Remark 3.1. This result is to be compared with Lemma 1.2 and Theorem 1.2. It shows that the addition of an initial layer improves the convergence result for small times since if t ∼ ε the first term in the rhs of (12) is of order 1.

Remark 3.2.
This result is useful since it shows that we found an approximation of the actual initial layer depending only on the data ζ 0 , β 0 and ρ I at t = 0.

Variation in time
Considering (2), ε multiplies ∂ t ρ ε . As we are interested in the convergence of ρ ε when ε goes to zero, one could ask where does ∂ t ρ ε belongs uniformly wrt ε and to which limit does it tend. To this aim we consider the system satisfied by D τ t ρ ε (the operator is defined as in (7)) : Theorem 3.2. Under hypotheses 1.1 and 1.2, and for every fixed τ small enough, one has : which gives the uniform bound in ε : where the constants C i are independent on ε, for i ∈ {1, 2, 3}.
Proof. One uses H, the functional introduced above and gets using Grönwall's Lemma that The main point of the proof is the control of the initial term H [D τ t ρ ε ( ·, 0)] as a function of ε and τ .
The first term decomposes in two parts where we used the method of characteristics. One splits the first term adding and subtracting intermediate terms ε As for I 1,2 , one has : where TV denotes total variation of ρ I [11]. In a similar way, as |ε∂ t µ 0,ε (t)| ≤ C, one obtains which ends the first part of the proof, then integrating (16) in time gives the other claim.

Now we concentrate on what remains near
We use the method of characteristics in order to express J 1 and J 2 as functions of initial and boundary values.

Results
The convergence results is calculated with a discrete formulation of (11) defined as: where u is a real sequence (u i ) i∈N .
Numerical results agree in a close manner to theoretical estimates stated in Theorems 1.2 and 3.1, for the L 1 and L ∞ norms either for ρ ε − ρ 0 or for the complete expansion ρ ε − ρ 0 −ρ 0,ε (cf Fig. 1). We see that even in the L 1 (0, T ) norm the complete zero order approximation is closer to ρ n ε,· although the convergence order is similar.

Perturbing the on and off rates
While in the second set of simulations, we perturb the data in order to test the accuracy of our estimates. β ε (t) = β min + (sin(2πt)) + + √ ε(cos(2kπt)) + ζ ε (a, t) = (1 + a)(1 + t) + √ ε(cos(2kπt)) + while the initial data is still ρ I (a) := exp(−a). When the initial layer is not part of the asymptotic expansion, the errors related to the layer dominate the perturbation. At the contrary, when the initial layer is included, the √ ε perturbation becomes perceptible in the error estimates (cf figs. 3 and 4).