A Cellular Potts Model of single cell migration in presence of durotaxis Rachele Allena, Marco Scianna, Luigi Preziosi To cite this version: Rachele Allena, Marco Scianna, Luigi Preziosi. A Cellular Potts Model of single cell migration in presence of durotaxis. Mathematical Biosciences, 2016, pp.57-70. hal-02375906 HAL Id: hal-02375906 https://hal.archives-ouvertes.fr/hal-02375906 Submitted on 22 Nov 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A Cellular Potts Model of single cell migration in presence of durotaxis ∗ R. Allena a, , M. Scianna b, L. Preziosi b a Arts et Metiers ParisTech, LBM/Institut de Biomecanique Humaine Georges Charpak, 151 bd de l’Hopital, 75013 Paris, France b Dipartimento di Scienze Mathematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy a b s t r a c t Cell migration is a fundamental biological phenomenon during which cells sense their surroundings and respond to different types of signals. In presence of durotaxis, cells preferentially crawl from soft to stiff substrates by reorganizing their cytoskeleton from an isotropic to an anisotropic distribution of actin fil- aments. In the present paper, we propose a Cellular Potts Model to simulate single cell migration over flat substrates with variable stiffness. We have tested five configurations: (i) a substrate including a soft and a stiff region, (ii) a soft substrate including two parallel stiff stripes, (iii) a substrate made of succes- sive stripes with increasing stiffness to create a gradient and (iv) a stiff substrate with four embedded soft squares. For each simulation, we have evaluated the morphology of the cell, the distance covered, the spreading area and the migration speed. We have then compared the numerical results to specific experimental observations showing a consistent agreement. Keywords: Cell migration Durotaxis Cell polarity Anisotropy CPM 1. Introduction Although several computational models have been proposed in literature to investigate single cell migration, only few of them Cell migration is a critical phenomenon occurring in several bi- deal with durotaxis. Among others, it is worth to cite the work ological processes, such as morphogenesis [1] , wound healing [2] by Moreo et al. [17] who proposed a continuum approach based and tumorogenesis [3] . It takes place in successive and cyclic steps on an extension of the Hill’s model for skeletal muscle behavior to [4] and it is triggered by specific interactions with the extracel- investigate cell response on two-dimensional (2D) substrates. They lular matrix (ECM). Actually, cell migration may occur in the ab- showed, in agreement with experimental observations, that cells sence of external signals thereby typically resulting in a random seem to have the same behavior when crawling on stiffer substrate walk. However, in most situations, cells are able to sense their and on pre-strained substrates. Harland et al. [18] instead repre- surrounding environment and to respond for instance to chemical sented a cell as a collection of stress fibers undergoing contraction (i.e., chemotaxis) [5] , electrical (i.e., electrotaxis) [6] or mechani- and birth/death processes and showed that on stiff substrates cells cal (i.e., mechanotaxis) [7] fields or yet to stiffness gradients (i.e., exhibit durotaxis and stress fibers significantly elongate. Dokukina durotaxis) [8,9] . The latter mechanism consists of the cell prefer- and Gracheva [19] developed a 2D discrete model of a viscoelas- ential crawling from soft matrix substrates to stiffer ones, even in tic fibroblast cell using a Delaunay triangulation. At each node the the absence of any additional directional cues [10,11] . By forming balance of the forces was determined by the contributions i) of local protrusions (i.e., pseudopodia), the cells are in fact able to the frictions between the cell and the substrate, ii) of a passive probe the mechanical properties of the surrounding environment viscoelastic force and iii) of an intrinsic active force. The authors and to more strongly adhere over stiff regions. Additionally, such then evaluated cell behavior over a substrate with a rigidity step behavior results in a substantial reorganization of the intracellu- in good agreement with specific experimental observations. In fact, lar cytoskeleton. In fact, over soft substrates cells typically show they found that the cell preferentially moves on the stiffer sub- an unstable and isotropic distribution of actin filaments, which are strate and turns away from the soft substrate as reported by [8] . poorly extended and radially oriented, whereas over stiff substrates Stefanoni et al. [20] proposed a finite element approach able to cell morphology is more stable and exhibits significant spreading account for the local mechanical properties of the underneath sub- and often anisotropic arrangements of actin filaments in the direc- strate and to analyze selected cell migratory determinants on two tion of migration (i.e., polarization) [12–16] . distinct configurations: an isotropic substrate and a biphasic sub- strate (which consists of two adjacent isotropic regions with dif- ferent mechanical properties). Trichet et al. [14] employed instead ∗ Corresponding author. Tel.: +33 1 44 24 61 18; fax: +33 1 44 24 63 66. the active gel theory to demonstrate that cells preferentially mi- E-mail address: [email protected] (R. Allena). grate over stiff substrates founding an optimal range of rigidity for http://dx.doi.org/10.1016/j.mbs.2016.02.011 a maximal efficiency of cell migration. Further, in [21] a vertex- The trial spin update is finally validated by a Boltzmann-like based approach (i.e., the so-called Subcellular Element Model, SCE) probability function defined as was set to represent intracellular cytoskeletal elements as well as − H T their mechanical properties. In particular, the dynamics of such P [ σ ( x source ) → σ ( x target )] (t ) = min 1 , e C (2) subcellular domains were described by Langevin equations, which ∈ account for a weak stochastic component (i.e., that mimic cyto- where t is the actual MCS and TC R+ is a Boltzmann temperature, plasmic fluctuations) and elastic responses (i.e., modeled by gen- that has been interpreted in several ways by CPM authors (see [33] eralized Morse potentials) to both intracellular and intercellular for a comment on this aspect). However, we here opt to give TC the biomechanical forces. The same method was successfully applied sense of a cell intrinsic motility (i.e., agitation rate), following the in [22] for modeling substrate-driven bacteria locomotion. Finally, approach in [25] . Finally, it is useful to underline that the matrix in Allena and Aubry [23] a 2D mechanical model was proposed to substrates are considered fixed and immutable. simulate cell migration over an heterogeneous substrate including As seen, the simulated system evolves to iteratively and slipping regions and to show that over softer regions the cell slows stochastically reduce its free energy, which is defined by a Hamil- down and is less efficient. tonian function H which, for any given time step t, reads In the present work, we describe a Cellular Potts Model (CPM, = + H(t ) Hadhesion (t ) Hshape (t ) (3) developed in [24,25] and reviewed in [25–29] ), which is a lattice- based stochastic approach employing an energy minimization phi- Hadhesion (t) is deduced from the Steinberg’s Differential Adhesion losophy, to reproduce single cell migration over flat substrates with Hypothesis (DAH) [24,34] and is due to the adhesion between cells different rigidity. In particular, we test four configurations: (i) a and extracellular components (i.e., the medium or a given type of substrate including a soft and a stiff region, (ii) a soft substrate substrate). In particular, it reads including two parallel stiff stripes, (iii) a substrate made of succes- H (t ) = H (t ) = J (4) adhesion adhesion τ ,τ ( ) ( σ ( x ) ) σ ( x ) sive stripes with increasing stiffness to create a gradient and (iv) ( ∂ x ∈ ∂ σ ) ∩ ( ∂ x ∈ ∂ ) a stiff substrate with four embedded soft squares. For each sce- σ nario, we analyze cell behavior in terms of morphology, distance with x and x two neighboring sites and σ and σ two covered, spreading/adhesive area and migration speed in order to neighboring objects (with borders ∂ σ and ∂ σ , respectively). capture the essential mechanisms of durotaxis. The computational J ∈ R + are constant and homogeneous binding τ ( σ ( ) ) ,τ ( σ ( ) ) outcomes are then compared with specific experimental observa- x x forces per unit area. They are symmetric with respect to their in- tions taken from the existing literature. dices and can be specified as follows: The rest of this paper is organized as follows. In Section 2 , we clarify the assumptions on which our approach is based and - J C,M is the adhesive strength between the cells and the col- present the model components. The simulation results are then lagenous medium which is constituted by a mixture of sol- shown in Section 3 . Finally, a justification of our model choices uble adhesive ligands (i.e., carbohydrate polymers and non- as well as a discussion on possible improvements is proposed in proteoglycan polysaccharides) and water solvent; Section 4 . Additionally, the article is equipped with an Appendix - J , gives the adhesive strength between the cells and i th type C Si that deals with statistics and parameter estimates. of substrate. Recalling the minimization theory of the CPM, we assume that the stiffer the substrate i, the lower the corre- sponding value J , (i.e., the higher the adhesion between the 2.