March 2015 EPL, 109 (2015) 58005 www.epljournal.org doi: 10.1209/0295-5075/109/58005 Tissue homeostasis: A tensile state N. Podewitz1,M.Delarue2 and J. Elgeti1 1 Theoretical Soft Matter and Biophysics, Institute of Complex Systems, Forschungszentrum Juelich 52425 Juelich, Germany 2 Physics Department, Stanley Hall, University of Berkeley - Berkeley, USA received 10 December 2014; accepted in final form 23 February 2015 published online 13 March 2015 PACS 87.19.R- –Mechanicalandelectricalpropertiesoftissuesandorgans PACS 87.17.Aa –Modeling,computersimulationofcellprocesses PACS 87.17.Ee –Growthanddivision Abstract –Mechanicsplayasignificantroleduringtissuedevelopment.Oneofthekeychar- acteristics that underlies this mechanical role is the homeostatic pressure, which is the pressure stalling growth. In this work, we explore the possibility of a negative bulk homeostatic pressure by means of a mesoscale simulation approach and experimental data of several cell lines. We show how different cell properties change the bulk homeostatic pressure, which could explain the benefit of some observed morphological changes during cancer progression. Furthermore, we study the dependence of growth on pressure and estimate the bulk homeostatic pressure of five cell lines. Four out of five result in a bulk homeostatic pressure in the order of minus one or two kPa. Copyright c EPLA, 2015 ⃝ Introduction. – While growth of eukaryotic cells is one of the walls acts as a movable piston connected to a mainly determined by signaling and nutrition, the notion spring. Cells are placed inside the chamber and the tis- that mechanics also play a role has been advancing con- sue grows, compressing the spring. Eventually, the force tinuously over the years. Ingber and colleagues noted exerted by the piston on the cells is large enough to slow already thirty years ago that tensile stresses can regu- down division, yielding a steady state, where apoptosis late tissue architecture [1]. Today, mechanotransduction balances cell division. and mechanobiology are important fields of research [2,3]. It is a challenging task to measure this homeostatic pres- From a physics point of view, a cell has to increase its sure experimentally. In some attempts, researchers em- volume in order to accommodate for new cell material. bedded cells in an elastic shell that can be deformed by In terms of thermodynamics, pressure is the conjugated the tissue, but the tissue never reached a steady state and variable of volume. Therefore, it seems natural to investi- therefore a well-defined pressure [7]. Another approach gate the pressure cells can develop as well as its feedback exerted a compressive stress directly onto multicellular onto growth. Mechanical feedback on growth has been spheroids [5,8,9], which resulted in a slow down in growth. implemented in many different ways [4,5]. One intuitive Adetailedanalysisshowedthatspheroidgrowthis approach is to expand the growth rate in powers of the highly dominated by surface effects in the sense that cell pressure around the zero growth rate pressure —the home- division occurs mainly at thefirst2–3celllayersofthe ostatic pressure [6]. In this theory, the homeostatic state free surface. Although some physiological gradients of nu- of a tissue is characterized by the balance of cell division trients or growth factors exist [10], this surface growth and cell death (apoptosis) rate, which reflects a dynamic effect is also seen in simulations [9], where we can de- steady state with respect to the number of cells. The tis- couple the mechanics from the biochemistry. A possible sue maintains a well-defined pressure, called homeostatic intuitive explanation by simple mechanics would be as fol- pressure PH ,andtheoverallgrowthratek around the lows. A growing cell increases its volume, creating a strain homeostatic state depends on the difference between PH dipole. The energy required to insert such a dipole at the and the externally imposed pressure P i: surface of the tissue is smaller than in the bulk as in the former case part of the necessary strain field is cut away. k = κ(P P i). (1) H − Therefore, division is favored at free surfaces. The concept is best understood with a simple gedankenex- The net growth in the bulk of the aforementioned tissue periment :atissueisconfinedinasmallchamber,where experiments was determined to be negative. This leads to 58005-p1 N. Podewitz et al. adynamicsteadystate,wheretheapoptoticcoreisbal- t =0d t =3d t = 15 d cell age [d] 0 anced by an influx of cells from the surface [11]. Could we thus deduce the homeostatic pressure to be negative? 0.2 Does a negative homeostatic pressure even make sense? On the one hand, many working in the field of tissue me- 0.4 chanics consider cells to be in a tensile state and, therefore, could indeed expect the homeostatic pressure to be nega- 0.6 tive. On the other hand, from a physical point of view, a F [a.u.] 0.8 material under tension seems to contradict expansion and | z| only yield unstable states. 0 3 15 t [d] We used the same mesoscopic simulation model as in [8,9,12] to perform a gedankenexperiment of tissue Fig. 1: (Color online) Visualization of the concept of a nega- growth in a box with a piston (see fig. 1 and Supplemen- tive homeostatic pressure. Sticky walls are modeled by parti- tary Movie S1 neghp.avi), demonstrating the concept cles similar to normal cells, except they are fixed with a strong and stability of a negative homeostatic pressure. Initially, harmonic potential (purple particles). The piston at the top is, cells are seeded at the bottom of the box, adhere to the furthermore, subjected to a weak harmonic potential depicted wall and start to grow upwards due to the free surface. by the spring. Snapshots at different times t are shown for The piston, acting as a sticky surface, however, cuts away the same simulation with (left) and without (right) piston (see this free surface and leaves only the on-average apoptotic Supplementary Movie S1 neghp.avi). Cells are color coded ac- bulk. The resulting decrease in cell number and the ad- cording to their age (red corresponds to recently divided cells, hesion forces between cells pull down the piston. Stress while blue corresponds to cells that did not divide for a certain in the bulk becomes tensile, and the growth rate increases amount of time). Note that most cell divisions occur at free surfaces. The sticky piston cuts away this free surface and is until a different steady state (under tension) is reached. pulled down due to the negative bulk homeostatic pressure. Methods. – In order to study the behavior of grow- ing tissues, we use the same mesoscopic simulation model is fixed, whereas the second derivative of the potential as in [8,9,12]. Briefly, each cell consists of two point par- around this minimum is varied, leading to softer/stiffer ticles that repel each other with a growth force F g = ij tissues. The compressibility K is defined as 1/f ∗. 2 0 B/(rij + r0) rˆij ,whereB is the growth force strength, In order to measure the bulk dependences on pressure, rij the distance between the particles i and j, r0 some we implement a constant pressure ensemble as detailed constant and rˆij the unit displacement between i and j. in [13]. The described method imposes a defined pressure Upon exceeding a certain critical distance, the cell divides. on a system with full periodic boundaries. This pressure is Apoptosis is introduced as a constant probability for each maintained through a periodic rescaling of the box volume cell to disappear. These two are the only active com- by a factor ponents in the simulations. The volume exclusion force ∆t i v 5 5 χ =1 βT (P P ), (2) Fij = f0 Rpp/rij 1 rˆij ensures impenetrability of the − tP − cells with the cell-cell− potential coefficient f .Allinterac- 0 where β is the isothermal compressibility, ∆t the sim- tions are! short ranged" with a cut-offradius R ,beyond T pp ulation time step, t arelaxationtimeconstantandP i which all interactions are set to zero. Cell adhesion is p the pressure we impose on the system (in simulations: represented by a simple constant force F a = f rˆ and ij 1 ij β /t =1d 1kPa 1 with a time step of ∆t =10 3 d). tuned by the adhesion strength f . − T P − − − 1 Furthermore, the center of mass of all cells is rescaled by To account for dissipation and random fluctuations, we √3 χ.Thereareingeneralnoconstraintsontheimposed use a dissipative particle dynamics (DPD)-type thermo- r R pressure, which makes it possible to simulate systems un- stat. It consists of a random force Fij = σω (rij )ξij rˆij der tension (P i < 0) as well. Following [14], the current with σ,thestrengthofthisforce,ξij = ξji asymmetric pressure P inside the full periodic simulation box is cal- Gaussian random variable with zero mean and unit vari- R culated through the evaluation of the virial stress ance and a weight function ω (rij )andadissipativeforce F d = γωD(r )(v rˆ )rˆ with γ the strength of this ij ij ij ij ij 1 force, v− = v v the· relative velocity between i and i i ij ij ij j i σαβ = mivαvβ + rα fβ , (3) − D −V ⎡ ⎤ j and a weight function ω (r ). Unless explicitly men- i i,j ij % % tioned, we use the same standard parameter set and nota- ⎣ ⎦ i tion as in [8,9], i.e. all quantities denoted with a ∗ are in where i is the sum over all particles, vα the α-th com- terms of the standard parameter set (e.g. B∗ = B/Bref). ponent of the velocity of particle i, i,j the sum over all ( ij i j To model different compressibilities, the volume exclu- pair interactions that act on i, rα = rα rα the α-th ( − ij sion force prefactor f0∗ and the adhesion force prefactor f1∗ component of the distance vector between i and j and fβ are varied simultaneously, while f0∗/f1∗ is kept constant.