# Stochastic Simulation of Pattern Formation in Growing Tissue: A Multilevel Approach

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## Abstract

We take up the challenge of designing realistic computational models of large interacting cell populations. The goal is essentially to bring Gillespie’s celebrated stochastic methodology to the level of an interacting population of cells. Specifically, we are interested in how the gold standard of single-cell computational modeling, here taken to be spatial stochastic reaction–diffusion models, may be efficiently coupled with a similar approach at the cell population level. Concretely, we target a recently proposed set of pathways for pattern formation involving Notch–Delta signaling mechanisms. These involve cell-to-cell communication as mediated both via direct membrane contact sites and via cellular protrusions. We explain how to simulate the process in growing tissue using a multilevel approach and we discuss implications for future development of the associated computational methods.

## Keywords

Reaction–diffusion master equation Discrete Laplacian cell mechanics Single-cell model Cell population model Notch signaling pathway## Mathematics Subject Classification

60J28 92-08 65C40## 1 Introduction

An important challenge in computational cell biology is to study the emergent behavior of single-cell pathways at the scale of a large interacting cell population. In this paper, we tackle this challenge by, in essence, attempting to generalize Gillespie’s stochastic simulation methodology to the level of the multicellular environment. In order to do so, clearly, the modeling physics of the extracellular space, of the cell population, and of the single cells need to be prescribed. A suitable computational methodology should additionally allow for cell-to-cell signaling in a flexible and general way. There are several possible interesting applications for such a kind of modeling framework, regulating processes in embryonic development, angiogenesis, neurogenesis, wound healing, and tumor growth, to mention just a few.

In the interest of focusing our work around a concrete, yet fairly demanding modeling situation, we pick as our target a specific set of network models which involve single-cell pathways together with non-trivial signaling between the individual cells. The Notch signaling pathway is a highly conserved mechanism which is present in most multicellular organisms (Artavanis-Tsakonas et al. 1999), ranging from, e.g., *Drosophila* and *C. elegans* to mammals. Indeed, the fundamental importance of Notch signaling made it an early target for mathematical models (Collier et al. 1996), where feedback regulation between neighbor cells was modeled. It has since been realized that cell-to-cell signaling not only is short range, taking place at direct junctional contact sites, but also is mediated via long range cellular protrusions (Cohen et al. 2010). Mathematical models including these effects have recently been investigated (Sprinzak et al. 2011; Hadjivasiliou et al. 2016), and we choose a family of such models as the concrete target in this paper.

To be able to realistically resolve the geometrical details of the single cell, unstructured meshes (e.g., triangularizations) stand out as a ubiquitous tool. Also, an important part of Dan Gillespie’s heritage to computational biology is that noisy cellular processes at the molecular level should be understood in a *stochastic* framework. These observations together suggest the *reaction–diffusion master equation* (RDME) over an unstructured mesh (Engblom et al. 2009), and we shall regard it herein as a gold standard in single-cell modeling. The RDME is based on first principles and is reasonably effective computationally. Additionally, this description, or simplified versions of it, has been successful at delivering important insights for a range of cellular phenomena (Fange and Elf 2006; Raj and van Oudenaarden 2008; Lestas et al. 2010; Barkai and Leibler 2000).

At the scale of a population of cells, cell-based computational modeling is an *in silico* approach to test hypotheses concerning the contributions of various mechanisms to observed macro-level behaviors. Examples of recent applications of cell-based models include embryonic development (Atwell et al. 2015), wound healing (Vermolen and Gefen 2013; Ziraldo et al. 2013), and tumor growth (Naumov et al. 2011). The natural analog of the RDME at the cell population level is found in the class of *on-lattice* cell-based models. As in the RDME, space is here discretized in a grid of voxels over which the cells are distributed. State update rules are then formulated over this grid where signaling processes and factor concentrations may be included via, e.g., differential equations (Robertson-Tessi et al. 2015).

In this work, we will focus on the novel on-lattice method proposed in Engblom et al. (2018), which is promising from a scaling point of view, yet also is very expressive. The method is referred to as *discrete Laplacian cell mechanics* (DLCM) and is formed by developing constitutive equations for the dynamics of the cell population at a given discretization of space. The update rules are stochastic and are established from global calculations. Importantly, the simulations take place in continuous time, thus allowing for a meaningful coupling to arbitrary continuous-time processes, including inter-cellular signals. In summary, we focus our attention to single-cell models described in the RDME framework and the main contribution of the paper is to investigate the feasibility of the two-level RDME-DLCM approach.

In the next section, we work through the specific, but fairly general, pattern forming mechanism we wish to study and we subsequently express it within the RDME-DLCM computational framework. As will be demonstrated, this enables us to simulate a range of intriguing patterns in an unprecedented detailed and bottom-up fashion. The paper is concluded with a discussion of some ideas concerning possible future developments of the presented computational methodology.

## 2 Models and Methods

Below we start by presenting the specific Notch pathway model, we will target in the paper. Throughout we consider a single non-dimensional model, originating from an attempt to map to the situation of explaining the organization of bristles on the *Drosphila* notum (Cohen et al. 2010). Such patterns are remarkably precise and are therefore good model systems to study the genetic basis of pattern formation. The model we decided to employ can be found in Hadjivasiliou et al. (2016). We make a slight extension of the model in Sect. 2.2 by bringing it into the spatial setting, essentially by deciding on a system volume and settling for suitable diffusion constants. In Sect. 2.3, we explain how to use the methodology in Engblom et al. (2018) to efficiently simulate a growing cell population. The two computational layers, i.e., the single-cell and the cell population layer, are put together in Sect. 2.4 where we present a few selected simulation results. In order to concentrate on the possibilities with the computational framework, we select spatial and cell population parameters rather freely, and we do not claim our resulting model to map to any specific real-world scenario.

### 2.1 Protrusion Mediated Notch–Delta Pattern Formation

*morphogens*, i.e., as done early on by Turing (1952), was based on lateral inhibition with feedback (Collier et al. 1996). This mechanism takes place in between the transmembrane proteins Notch and Delta, respectively. In a non-dimensional setting, with \((n_i,d_i)\) denoting the Notch- and Delta concentrations within cell

*i*, the original model has the form (Collier et al. 1996)

*i*. In (1),

*f*and

*g*denote monotonically increasing and decreasing functions of their single argument, respectively.

*Drosophila*, communication of Delta via cellular protrusions was added to the model in Cohen et al. (2010). Later details were added in Sprinzak et al. (2011), Hadjivasiliou et al. (2016), including differential weighting of the incoming signals and an addition of the concentration of a Notch reporter molecule \(r_i\). More specifically, the model from Hadjivasiliou et al. (2016) reads

*a*) and (

*b*) denote the sum of the signal over all cells making

*junctional*and, respectively,

*protrusional*contact (see Fig. 1). More concretely,

*J*(

*i*) and

*P*(

*i*) denote the set of junctional and protrusional contacts for cell

*i*. A specific novelty with this model is \(\langle d_{rm out} \rangle \), the total amount of bound Delta that leads to activation of the Notch receptor. Differential weighting of the signals is achieved by assuming different constant weights \([w_a,w_b,q_a,q_b]\) of the incoming signals.

*n*,

*d*,

*r*) in cell

*i*as absolute molecular counts (

*N*,

*D*,

*R*) at some fix system volume \(\Omega \). From (2), we propose the transitions

Simulation results for this model when starting from a random initial configuration are summarized in Fig. 2 at a system volume \(\Omega = 400\). We now proceed to make an immediate spatial extension of this model.

### 2.2 Spatial Stochastic Reaction–Diffusion Models of Single Cells

Living cells are inherently inhomogeneous objects, and the assumption of well-stirredness can rightly be questioned (Dobrzyński et al. 2007; Fange and Elf 2006). The reaction–diffusion master equation (RDME) attempts to strike a balance between accuracy and computational efficiency (Gardiner 2004). Here the domain under consideration is discretized in small enough compartments, or *voxels*, such that diffusion is enough to regard each voxel as well stirred. Diffusion in between voxels is handled as a special set of reactions with rates obtained so as to match with macroscopic diffusion properties (Engblom et al. 2009). An efficient algorithm for spatial stochastic simulation is the Next subvolume method (NSM) (Fange and Elf 2006), which can be thought of as a blend of Gillespie’s Direct method with the Next reaction method. The algorithm is summarized in “Appendix A.”

To illustrate the single cell population-level approach we have in mind, we make an immediate version of pathway model (5) as follows. As a single-cell discretization, we take the triangularization depicted in Fig. 3 which consists of a modest number of 40 voxels. We make no particular distinction of the membrane, the cytoplasm, or the nuclei, but allow all reactions in (5) to take place in all of the voxels. We let the geometry be of total volume \(\Omega = 400\) and use the scalar diffusion constant \(1/\Omega \) across the whole cell geometry and for all species [*N*, *D*, *R*]. Although there are clearly many potential improvements to this basic model, it will serve as an interesting load case to our simulation approach. We thus have to postpone for another occasion the interesting quest for additional modeling realism including, e.g., nuclei- and membrane specific transitions.

### 2.3 Stochastic Simulation of Growing Cell Populations

Given the relative efficiency of the RDME approach, one can wonder if not a similar idea could be useful at the cell population level. Unlike the various molecules inside the living cell, however, cells in multicellular structures do not generally diffuse around freely. Instead, cells may actively crawl, adhere to other cells, and are pushed into position. An RDME-like framework for this situation was recently developed, and we now briefly review this idea (Engblom et al. 2018).

*i*and let \((p_i)\) be the corresponding cellular pressure. Denote by \(\Omega _h\) the subset of voxels \(v_i\) for which \(u_i \not = 0\) and let \(\partial \Omega _h\) denote the discrete boundary, the set of unpopulated voxels that share an edge with a voxel in \(\Omega _h\). At any instant in time

*t*, we solve for the cellular pressure,

*L*is a consistent discretization of \(\Delta \) over \(\Omega _h\) and where the source term is \(s(u_i) = 0\) for \(u_i \le 1\) and \(s(u_i) = 1\) whenever \(u_i = 2\). This normalization ensures that \(p = 0\) at equilibrium. It is doable to rely on this setup also for unstructured meshes by postulating that the cellular pressure is proportional to the difference in volume occupancy and voxel volume. However, there are biological specifics which should rightly be considered in this case, such as adhesion effects in voxels populated under their carrying capacity, and also details concerning the volume characteristics of the individual cells.

*rate*for the event that one cell moves from voxel

*i*to

*j*is taken to be

*D*may depend on position and on the type of movement. We take \(D = 0\) for movements into voxels containing an equal number of cells, thus limiting the cellular movements to less crowded voxels only.

We exemplify the process by growing a small population of 1000 cells, starting from a single cell and allowing it to proliferate at a certain rate provided it has enough concentration of “nutrition.” The nutrition is distributed at the boundaries \(\Omega _h\) of the cell population, and we let it diffuse by the Laplace operator. At any given time, cells consume nutrition for their own metabolism and so this scheme will favor the proliferation of cells near the boundary where the nutrition concentration is the highest (see Fig. 5). In the next section, we proceed by coupling this DLCM layer growth process to the previously developed RDME-layer description of the Notch–Delta pattern formation mechanism. Hence, the fine RDME discretization as depicted in Fig. 3 is used to describe the physics of the individual cell, whereas the DLCM grid is used for the cell population.

### 2.4 A Range of Notch–Delta Patterns in Growing Tissue

In Algorithm 2.1, we expand the details of the inner layer simulation where the discretization in time chunks \([t,t+d\tau )\) is made explicit. It follows that the assumption made here is essentially that the Notch–Delta dynamics takes place on a faster timescale than the growth process, a quite reasonable assumption in this case. Without this assumption, the simulation efficiency will deteriorate whenever \(d\tau \) for accuracy reasons has to be chosen small.

## 3 Conclusions

The main focus of this paper has been to investigate the feasibility of a two-level RDME-DLCM approach. We choose the RDME description of a single cell as a gold standard modeling approach. This is a detailed, flexible, yet also comparably effective simulation methodology. At the cell population level, the related DLCM method was used and the two layers of description were coupled together with relative ease.

The main reason this combination is convenient is the fact that both layers take place in continuous time and can be simulated by Gillespie-style event-driven algorithms. We also point out that the overall method combination is promising from the point of view of deriving approximate simulation algorithms, as, for example, shown in detail for the RDME framework in Chevallier and Engblom (2018). A concrete example is that, for practical reasons in the implementation, we had to discretize time for the cell-to-cell signaling process of the model, cf. (7). Although not discussed here, one can expect that this method has a strong error of order \(O(d\tau ^{1/2})\) Engblom (2015). Since we selected a quite conservative time step in (7), we believe that our implementation is a bit inefficient in that the time-step restriction is too restrictive given the accuracy demands of the application at hand. This is an issue which could clearly be of interest to target in future research toward faster algorithms. Other related ideas are deterministic-stochastic hybrid algorithms Chevallier and Engblom (2018), Haseltine and Rawlings (2002), Lo et al. (2016) and, more generally, multiscale solvers based on ideas from stochastic homogenization techniques (Ea et al. 2005; Cao et al. 2005b, a; Cao and Petzold 2008). At the DLCM layer, the computational bottleneck lies in factorizing the Laplacian operator. Real savings in computing time here can be expected from employing traditional multigrid techniques Stüben (2001), Trotter et al. (2001).

Lastly but not the least, the computational framework described clearly opens up for many interesting applications where the emerging cell population behavior of detailed whole-cell models is to be approached.

### 3.1 Availability and Reproducibility

The computational results can be reproduced within the upcoming release 1.4 of the URDME open-source simulation framework Drawert et al. (2012), available for download at www.urdme.org.

## Notes

### Acknowledgements

Zena Hadjivasiliou kindly and patiently detailed several aspects of the running model used throughout this paper (Hadjivasiliou et al. 2016).

## Supplementary material

## References

- Andrews SS, Bray D (2004) Stochastic simulation of chemical reactions with spatial resolution and single molecule detail. Phys Biol 1(3):137–151. https://doi.org/10.1088/1478-3967/1/3/001 CrossRefGoogle Scholar
- Artavanis-Tsakonas S, Rand MD, Lake RJ (1999) Notch signaling: cell fate control and signal integration in development. Science 284(5415):770–776. https://doi.org/10.1126/science.284.5415.770 CrossRefGoogle Scholar
- Atwell K, Qin Z, Gavaghan D, Kugler H, Hubbard EJA, Osborne JM (2015) Mechano-logical model of C. elegans germ line suggests feedback on the cell cycle. Development 142(22):3902. https://doi.org/10.1242/dev.126359 CrossRefGoogle Scholar
- Barkai N, Leibler S (2000) Circadian clocks limited by noise. Nature 403:267–268. https://doi.org/10.1038/35002258 CrossRefGoogle Scholar
- Cao Y, Petzold LR (2008) Slow scale tau-leaping method. Comput Methods Appl Mech Eng 197(43):3472–3479. https://doi.org/10.1016/j.cma.2008.02.024 MathSciNetCrossRefzbMATHGoogle Scholar
- Cao Y, Gillespie DT, Petzold LR (2005a) The slow-scale stochastic simulation algorithm. J Chem Phys 122(1):014116. https://doi.org/10.1063/1.1824902
- Cao Y, Gillespie DT, Petzold LR (2005b) Multiscale stochastic simulation algorithm with stochastic partial equilibrium assumption for chemically reacting systems. J Comput Phys 206:395–411. https://doi.org/10.1016/j.jcp.2004.12.014
- Chevallier A, Engblom S (2018) Pathwise error bounds in multiscale variable splitting methods for spatial stochastic kinetics. SIAM J Numer Anal 56(1):469–498. https://doi.org/10.1137/16M1083086 MathSciNetCrossRefzbMATHGoogle Scholar
- Cohen M, Georgiou M, Stevenson NL, Miodownik M, Baum B (2010) Dynamic filopodia transmit intermittent delta-notch signaling to drive pattern refinement during lateral inhibition. Dev Cell 19(1):78–89. https://doi.org/10.1016/j.devcel.2010.06.006 CrossRefGoogle Scholar
- Collier JR, Monk NA, Maini PK, Lewis JH (1996) Pattern formation by lateral inhibition with feedback: a mathematical model of Delta-Notch intercellular signalling. J Theor Biol 183(4):429–446. https://doi.org/10.1006/jtbi.1996.0233 CrossRefGoogle Scholar
- Dobrzyński M, Rodríguez JV, Kaandorp JA, Blom JG (2007) Computational methods for diffusion-influenced biochemical reactions. Bioinformatics 23(15):1969–1977. https://doi.org/10.1093/bioinformatics/btm278 CrossRefGoogle Scholar
- Drawert B, Engblom S, Hellander A (2012) URDME: a modular framework for stochastic simulation of reaction-transport processes in complex geometries. BMC Syst Biol 6(76):1–17. https://doi.org/10.1186/1752-0509-6-76 Google Scholar
- Ea W, Liu D, Vanden-Eijnden E (2005) Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J Chem Phys 123(19):194107. https://doi.org/10.1063/1.2109987 CrossRefGoogle Scholar
- Engblom S (2015) Strong convergence for split-step methods in stochastic jump kinetics. SIAM J Numer Anal 53(6):2655–2676. https://doi.org/10.1137/141000841 MathSciNetCrossRefzbMATHGoogle Scholar
- Engblom S, Wilson D, Baker R (2018) Scalable population-level modeling of biological cells incorporating mechanics and kinetics in continuous time. Accepted for publication in R Soc Open Sci. arxiv:1706.03375
- Engblom S, Ferm L, Hellander A, Lötstedt P (2009) Simulation of stochastic reaction–diffusion processes on unstructured meshes. SIAM J Sci Comput 31(3):1774–1797. https://doi.org/10.1137/080721388 MathSciNetCrossRefzbMATHGoogle Scholar
- Fange D, Elf J (2006) Noise-induced Min phenotypes in \(E. coli\). PLoS Comput Biol 2(6):637–648. https://doi.org/10.1371/journal.pcbi.0020080 CrossRefGoogle Scholar
- Gardiner CW (2004) Handbook of stochastic methods. Springer series in synergetics, 3rd edn. Springer, BerlinCrossRefGoogle Scholar
- Hadjivasiliou Z, Hunter GL, Baum B (2016) A new mechanism for spatial pattern formation via lateral and protrusion-mediated lateral signalling. J R Soc Interface 13(124):1–10. https://doi.org/10.1098/rsif.2016.0484 CrossRefGoogle Scholar
- Haseltine EL, Rawlings JB (2002) Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics. J Chem Phys 117(15):6959–6969. https://doi.org/10.1063/1.1505860 CrossRefGoogle Scholar
- Lestas I, Vinnicombe G, Paulsson J (2010) Fundamental limits on the suppression of molecular fluctuations. Nature 467(7312):174–178. https://doi.org/10.1038/nature09333 CrossRefGoogle Scholar
- Lo W-C, Zheng L, Nie Q (2016) A hybrid continuous-discrete method for stochastic reaction–diffusion processes. R Soc Open Sci 3(9). https://doi.org/10.1098/rsos.160485
- Naumov L, Hoekstra A, Sloot P (2011) Cellular automata models of tumour natural shrinkage. Physica A 390(12):2283–2290. https://doi.org/10.1016/j.physa.2011.02.006 CrossRefGoogle Scholar
- Puchalka J, Kierzek AM (2004) Bridging the gap between stochastic and deterministic regimes in the kinetic simulations of the biochemical reaction networks. Biophys J 86(3):1357–1372. https://doi.org/10.1016/S0006-3495(04)74207-1 CrossRefGoogle Scholar
- Raj A, van Oudenaarden A (2008) Nature, nurture, or chance: stochastic gene expression and its consequences. Cell 135(2):216–226. https://doi.org/10.1016/j.cell.2008.09.050 CrossRefGoogle Scholar
- Robertson-Tessi M, Gillies RJ, Gatenby RA, Anderson AR (2015) Impact of metabolic heterogeneity on tumor growth, invasion, and treatment outcomes. Cancer Res 75(8):1567–1579. https://doi.org/10.1158/0008-5472.CAN-14-1428 CrossRefGoogle Scholar
- Shimoni Y, Nudelman G, Hayot F, Sealfon SC (2011) Multi-scale stochastic simulation of diffusion-coupled agents and its application to cell culture simulation. PLoS ONE 6(12):1–9. https://doi.org/10.1371/journal.pone.0029298 CrossRefGoogle Scholar
- Sprinzak D, Lakhanpal A, LeBon L, Garcia-Ojalvo J, Elowitz MB (2011) Mutual inactivation of notch receptors and ligands facilitates developmental patterning. J R Soc Interface 7(6):1–11. https://doi.org/10.1371/journal.pcbi.1002069 MathSciNetGoogle Scholar
- Stüben K (2001) A review of algebraic multigrid. J Comput Appl Math 128(1–2):281–309. Numerical analysis 2000. Partial differential equations, vol VII. https://doi.org/10.1016/S0377-0427(00)00516-1
- Trotter U, Oosterlee CW, Shüller A (2001) Multigrid. Academic Press, LondonGoogle Scholar
- Turing AM (1952) The chemical basis of morphogenesis. Philos Trans R Soc Lond B 237(641):37–72. https://doi.org/10.1098/rstb.1952.0012 MathSciNetCrossRefzbMATHGoogle Scholar
- Vermolen FJ, Gefen A (2013) A semi-stochastic cell-based model for in vitro infected ‘wound’ healing through motility reduction: a simulation study. J Theor Biol 318:68–80. https://doi.org/10.1016/j.jtbi.2012.11.007 MathSciNetCrossRefzbMATHGoogle Scholar
- Ziraldo C, Mi Q, An G, Vodovotz Y (2013) Computational modeling of inflammation and wound healing. Adv Wound Care 2(9):527–537. https://doi.org/10.1089/wound.2012.0416 CrossRefGoogle Scholar

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