# A 3D Multiscale Model to Explore the Role of EGFR Overexpression in Tumourigenesis

## Abstract

The epidermal growth factor receptor (EGFR) signalling cascade is one of the main pathways that regulate the survival and division of mammalian cells. It is also one of the most altered transduction pathways in cancer. Acquired mutations in the EGFR/ERK pathway can cause the overexpression of EGFR on the surface of the cell, while others downregulate the inactivation of switched on intracellular proteins such as Ras and Raf. This upregulates the activity of ERK and promotes cell division. We develop a 3D multiscale model to explore the role of EGFR overexpression on tumour initiation. In this model, cells are described as individual objects that move, interact, divide, proliferate, and die by apoptosis. We use Brownian Dynamics to describe the extracellular and intracellular regulations of cells as well as the spatial and stochastic effects influencing them. The fate of each cell depends on the number of active transcription factors in the nucleus. We use numerical simulations to investigate the individual and combined effects of mutations on the intracellular regulation of individual cells. Next, we show that the distance between active receptors increase the level of EGFR/ERK signalling. We demonstrate the usefulness of the model by quantifying the impact of mutational alterations in the EGFR/ERK pathway on the growth rate of in silico tumours.

## Keywords

EGFR Tumour growth Agent-based modelling Brownian Dynamics## 1 Introduction

The EGFR/ERK cascade is one of the main mitogen activated-protein kinase (MAPK) pathways that regulate the survival and division of several mammalian cells (Orton et al. 2005). This signalling cascade is stimulated when epidermal growth factors (EGFs) bind to their receptors (EGFRs) present on the membrane surface of cells. As a result, Ras proteins, which diffuse in proximity of the corresponding G-protein site, bind to GTP molecules and switch to the Ras-GTP active state. The diffusing Ras-GTPs subsequently interact with Raf which, upon activation, dually phosphorylate and activate MEK proteins. Subsequently, MEKs participate in the formation of active ERKs. Then, ERK proteins phosphorylates RSK and both of them translocate to the nucleus where they activate several transcription factors such as CREB, Fos, and Elk-1.

In cancer, acquired mutations disrupt the normal functioning of the EGFR/ERK pathway. Some of these mutations lead to the upregulation of ERK activation which promotes the survival and proliferation of the cell. The most clinically observed mutations of the EGFR/ERK pathway concern changes in the Ras and Raf oncogenes as well EGFR. In this context, it was observed that 30% of human cancers contain a mutation in the Ras gene (K-Ras, H-Ras, N-Ras) (Bos 1989). These mutations prevent the automatic switching off of the Ras protein upon its activation. Another type of mutation that is often observed in human cancers concerns the B-Raf oncogene (Davies et al. 2002). This mutation leads to the production of permanently activated Raf proteins.

Alterations in the EGFR gene represent another type of the commonly observed cancerous mutations. These changes can provoke the overexpression of EGFR proteins by the cell. EGFR mutations are usually observed in lung cancer as more than 60% of the non-small lung carcinoma cells overexpress EGFRs (da Cunha Santos et al. 2011). There exist several cancer treatments that target the different components of the EGFR/ERK pathway (Roberts and Der 2007). For example, EGFR inhibitors are a class of these treatments that directly block the epidermal growth factor receptors and prevent their activation. Other drugs used in cancer treatment infiltrate the tumour cells and inhibit the activation of some proteins in the EGFR/ERK cascade. Overall, a proper understanding of the effects of mutations on cells is of paramount importance to design more effective anti-cancer therapeutic strategies.

Due to its importance, the EGFR/ERK pathway has been extensively studied experimentally and computationally in previous works. The dynamics of the EGF-receptor were studied using a variety of modelling techniques (Wiley et al. 2003). Detailed simulation of the EGFR/ERK cascade signalling was possible using ordinary differential equations (ODEs). For example, one of the developed models that describe the EGFR/ERK pathway consists of 94 state variables and 95 parameters (Schoeberl et al. 2002). Other ODE-based models quantified the effects of cancerous mutations on the kinetics of the EGFR/ERK cascade upon the activation of EGF receptors (Brown et al. 2004; Orton et al. 2009).

Several modelling techniques and methods were previously used to describe tumour development. The evolution of the populations of tumour cells were often represented using a continuous approach. In this context, both ordinary differential equations and partial differential equations (PDEs) were used to build mathematical models that study several questions related to the organization of tumours (Byrne and Chaplain 1996; Glass 1973; Wise et al. 2008). For example, it was possible to use continuous models to investigate other important questions in oncology such as the intraclonal heterogeneity of tumours (Stiehl et al. 2014; Walenda et al. 2014) and its role in chemotherapy resistance (Panetta 1998). The implementation of continuous models is straightforward, and it is possible to analyse them analytically in order to derive key insights on the underlying mechanisms that govern tumour growth. However, they do not accurately capture cell–cell and/or receptor–ligand interactions. For this reason, discrete models, where cells are represented as individual agents, are often formulated to describe cancer systems. For example, cellular automata (CA) models were previously used to simulate the self-organization of tumour cells (Aubert et al. 2006). CA, as well as the other discrete modelling techniques, describes cells as individual objects that move, interact, divide, and die. Discrete models can be either on-lattice or off-lattice. These models focus on the interactions between cells, but they do not capture the intracellular and extracellular mechanisms underlying the regulation of cell fate. In reality, complex physiological systems, such as cancerous tumours, are regulated by a wide range of processes taking place at different scales of space and time. In this context, multiscale models seek to combine the previously described models in order to benefit from their advantages and overcome their shortcomings.

Hybrid discrete-continuous models, which can be also have a multiscale organization, have been applied successfully to simulate tumour growth (Ramis-Conde et al. 2008, 2009; Zhang et al. 2009). In these models, cells are represented as discrete objects whose intracellular and extracellular regulation are described by continuous models (ODEs and PDEs). We have previously used this modelling technique to describe the development of multiple myeloma, its intraclonal heterogeneity, and effect on erythropoiesis (Bouchnita et al. 2016a, 2017a). Yet, these hybrid models do not capture the spatial and stochastic effects that shape the regulation of cells. Indeed, a key insight from the last decade of systems biology research is that stochastic models of chemical kinetics can capture and predict effects from stochasticity in gene regulation caused by low molecular copy numbers. Recently, a range of studies have highlighted how the spatial dimension of intracellular regulation plays a key role in the functioning of the underlying systems (Elf and Ehrenberg 2004; Fange and Elf 2016; Lawson et al. 2013; Sturrock et al. 2013b, a; van Zon et al. 2006; You et al. 2016).

The majority of the previously developed multiscale models of tumour growth relies on a deterministic description for the intracellular and extracellular regulatory processes to reduce the computational cost. Furthermore, most of them focus on the higher-level physiological processes such as cell–cell and cell–microenvironment interactions. However, acquired cancerous mutations alter the normal functioning of cellular components such as receptors and proteins involved in the regulation of the cell. In this work, we present a novel multiscale model that combines three modules describing physiological processes that regulate tumour growth at different scales. These modules include the interaction between EGFRs and their ligands, the intracellular regulation of individual cells via the EGFR/ERK pathway, and the 3D multicellular biomechanics of early-stage tumour growth. We represent cells as discrete objects that can move, grow, divide, and die by apoptosis. Each cell is modelled as a sphere with a nucleus, a cytoplasm where different molecules diffuse, and a membrane with surface receptors. The interaction between receptors and their extracellular ligands (EGFs) is described with Brownian Dynamics. The same method is used to simulate the dynamics of intracellular regulation. The fate of each cell depends on the number of activated transcription factors in its nucleus. Thus, the model integrates a detailed spatial model of the intracellular regulation following extracellular signalling, and tumour biomechanics. The fidelity of the model makes it suitable to properly study the fine grained aspects of tumour initiation and early-stage growth. We first apply the model to study the effects of acquired mutations in the EGFR/ERK pathway on the single-cell dynamics and quantify the impact of the distance between active EGFRs on the fate of individual cells. Then, we describe the pathogenesis of the 3D in silico tumours and the effect of mutational changes on their growth rate.

The remainder of this paper is organized as follows: Sect. 2 introduces the multiscale model of tumour growth, including the different modules and the techniques used to implement them. Section 3 presents the results of numerical simulations quantifying the effects of mutational alterations on the intracellular regulation of individual cells and their impact on tumourigenesis. The novelty, contributions, and limitations of this study are discussed in Sect. 4.

## 2 A Multiscale Model to Simulate the Effect of EGFR/ERK Signal Transduction on 3D Tumour Growth

### 2.1 Extracellular Regulation of Cells Through EGFR–Ligand Interactions

### 2.2 Intracellular Regulation

To capture the effects of mutational alterations on the fate of each cell, we develop a submodel of intracellular regulation via EGFR/ERK signalling. We use the Brownian Dynamics method to implement the model in order to preserve the spatial and stochastic effects underlying this process. A snapshot of the structure of an individual cell is presented in Fig. 2b.

*Modelling the EGFR/ERK signalling pathway*

EGFR/ERK signalling begins when EGFRs form G-protein binding domains upon their activation by EGF particles. These domains activate the neighbouring Ras proteins which results in the induction of the Ras/Raf/MEK/ERK cascade. The part of the EGFR/ERK pathway described by the model is shown in Fig. 3. We consider the following reactions:

*Brownian Dynamics Simulation*

We implement the previously described model of EGFR/ERK signalling using the BD method. Molecules in the cytoplasm are modelled as hard spheres diffusing according to Brownian motion. Each molecule has a reaction radius \(r_\mathrm{BD}\). If two reactive molecules overlap, the corresponding reaction fires. The activation duration of proteins is sampled from an exponential distribution. Acquired mutations in the Ras and Raf proteins reduce their corresponding inactivation rates. We consider that intracellular proteins do not degrade during the lifetime of the cell. This assumption was considered because the half-life time of these proteins is about 24 h which corresponds to the average lifetime of the cell Ramalho et al. (2002).

The numbers and diffusion domains of molecular species involved in the model

Molecular species | Number in the model | Copy number in the experiments | Diffusion domain |
---|---|---|---|

Ras | 200 | 400,000 (estimated from Orton et al. (2009)) | Cytoplasm |

Raf | 60 | 120,000 (estimated from Orton et al. (2009)) | Cytoplasm |

ERK | 300 | 600,000 (estimated from Orton et al. (2009)) | Whole cell |

TF | 50 | 100,000 (estimated from Biggin (2011)) | Nucleus |

The extracellular and intracellular molecules are propagated using an operator split approach. We choose a time step \(\Delta t\), propagate the extracellular molecules, then the intracellular molecules, and finally the action of the cells. At the end of the time step, the system is synchronized.

There exist a variety of microscale simulators. Examples are Smoldyn (Andrews and Bray 2004), MCell (De Schutter 2000), and eGFRD (van Zon and Ten Wolde 2005). The former two implement similar fixed time step BD schemes, while the latter, eGFRD, is more detailed and accurate. To simplify the integration with the cell mechanics model, we implemented a BD solver specific to the model that we study in this work. Also, using a more detailed and accurate microscale method, such as eGFRD, would make the simulations prohibitively expensive, and would not be warranted given the resolution of the model.

Simulations can be significantly speeded up compared to a naive BD scheme by considering that molecules in the extracellular space are simulated independently from molecules in the intracellular space. In addition, we do not account for interactions between non-reactive molecules. Finally, we use a relatively large time step, as, in the present study, the qualitative behaviour of the system is more important than the quantitative.

### 2.3 A Cell-Based Model of Tumour Growth Mechanics

*m*is the mass of the particle and \(\mu \) is the friction factor due to contact with the surrounding medium. The repulsive force between two cells is given explicitly by:

*i*and

*j*, \(h_0\) is the sum of their radii,

*K*is a positive parameter, and \(h_1\) is the sum of the incompressible part of each cell. The force between the particles tends to infinity if \(h_{ij}\) decreases to \(h_0 - h_1\). The same modelling method was previously used to describe the biomechanics of multicellular systems such as erythropoiesis (Bouchnita et al. 2016b), multiple myeloma (Bouchnita et al. 2016a, 2017a), and the immune-response (Bouchnita et al. 2017b, c). This cell-based model describes the biomechanics of tumour growth in environments with different Reynolds numbers. It can be reduced to a first-order equation by assuming that the inertial term is extremely small in comparison with the dissipative term (Galle et al. 2005; Ghaffarizadeh et al. 2018; Letort et al. 2018; Macklin et al. 2012; Pitt-Francis et al. 2009). This assumption is especially valid for tumours that grow in microenvironments with a low Reynolds number. In our model, we keep the inertia term because we would like to simulate tumour development in different locations of the body.

### 2.4 Sources of Noise in the Model

- 1.
the distribution of G-protein binding sites on the cell membrane;

- 2.
the stochastic residence time of EGFRs;

- 3.
the low copy number of intracellular molecular species.

- 1.
the relatively low number of EGF particles and their random introduction to the domain as well as stochasticity in their degradation time;

- 2.
the fluctuations in the cell cycle;

- 3.
the low number of cells per simulation;

- 4.
the dilution of proteins at the moment of cell division.

## 3 Results

### 3.1 The Effect of Mutational Alterations in the EGFR/ERK on the Intracellular Regulation of Cells

We illustrate the performance of the model by investigating the effect of cancerous mutations in the EGFR/ERK pathway on the regulation of individual cells as described in a previous work (Brown et al. 2004). This previous study quantified the impact of the K-Ras and B-Raf mutations on the activation of ERK proteins. To achieve this, a validated ODE model was developed and used to describe the case where an active receptor stimulates the EGFR/ERK pathway. The receptor is considered to be activated during the first 8 min. It becomes inactive after this time. As a result, the activation of ERK proteins in the absence of mutations is transient in the normal case. However, when there are additional K-Ras or B-Raf mutations, the observed ERK activation becomes constitutive. The same behaviour was observed in another study on the intracellular activity of ERK (Orton et al. 2009).

We calibrate our model for one cell to reproduce these findings. We consider a cell consisting of intracellular proteins all initially in the ‘off’ state and we set the number of initially active receptor clusters to one. We consider that this receptor becomes inactive after 8 min which is equal to the average of the waiting time of EGFR inactivation. First, we only consider the dynamics of the EGFR/ERK pathway in normal conditions. Then, we simulate the intracellular regulation dynamics when acquired K-Ras and B-Raf mutations upregulate the activity of ERK. These mutations decrease the inactivation rates of the corresponding proteins.

### 3.2 The Distance Between Simultaneously Active EGFRs Affects the Level of EGFR/ERK Signalling

The activation of EGFRs results in the apparition of G-protein binding sites at the inner surface of the cell membrane. These sites stimulate the EGFR/ERK signalling pathway which promotes the survival and proliferation of cells. Non-spatial models can be used to quantify the effects of EGFR–ligand binding on the intracellular regulation of cells. However, these models do not consider the spatial distribution of EGFRs at the surface of the cell and its effect on cellular regulation. Our model can describe the effect of the spatial distribution of receptors on the intracellular regulation of cells.

### 3.3 The Effect of EGFR Overexpression on Tumourigenesis

### 3.4 The Effect of Mutations on the Rate of Tumour Growth

## 4 Discussion

The main purpose of this work was to develop a high-fidelity model and simulation framework capable of demonstrating that detailed spatial stochastic aspects of gene regulation can interplay with cell mechanics in the early stages of tumour initiation. Even with state-of-the-art methodology, such simulations become highly computationally demanding. To make simulations feasible, simplifications had to be introduced in the Brownian Dynamics particle simulations to reduce the computational time. These simplifications include the reduction in the number of proteins and the simplification of reaction models below the physiologically observed copy numbers of the precise proteins in the network that we consider. This means precise quantitative predictions are out of scope of the present model, and in particular, it could risk to exaggerate the relative protein noise levels for the particular intracellular regulation network in this paper. However, the framework serves the purpose of qualitatively exploring how detailed spatial stochastic kinetics and mutational alterations in single cells can influence phenomenological predictions. Our numerical experiments clearly illustrate a number of processes that are possible to study when stochastic single-cell kinetics is incorporated in a cell-based framework. Encouraged by these predictions, we are actively pursuing a) multiscale techniques to reduce the cost of single-cell simulations to extend the range of protein copy numbers that can be feasibly simulated, and b) massively parallel simulations in modern distributed and cloud environments using the Orchestral toolkit (Coulier and Hellander 2018). Indeed, an efficient computer implementation is necessary in order to perform numerical simulations of complex tumour models in an affordable CPU time (Szymańska et al. 2018). Therefore, we wrote our code in a modular way and in the object-oriented programming (OOP) style. The components of the model, such as cells, receptors, cytokines, and proteins, were introduced as classes in the code. The computational space was divided into boxes to efficiently simulate the interactions between proteins and cells. To further reduce the computational cost of BD simulations, we intend to implement a tau-leaping method in the future (Gillespie 2001).

The reported multiscale model represents a novel tool to study the dynamics of tumour initiation under various conditions. The model is organized in a multiscale architecture which integrates several modules for the underlying physiological processes ranging from the intracellular dynamics of individual cells and the EGFR–ligands interactions to the biomechanics of multicellular systems. The performance of our framework was tested using a bottom-top approach. We began by investigating the effects of mutational alterations on the intracellular dynamics of individual cells. The predictions of the model qualitatively agree with the findings reported in earlier work where the kinetics of the EGFR/EERK pathway were simulated (Brown et al. 2004; Orton et al. 2009). The effects of the spatial distribution of active receptors on the regulation of individual cells were also explored using numerical simulations. Then, we have quantified the individual and combined effects of mutations on the dynamics of tumour initiation. Despite the low number of considered cells and the various sources of stochastic noise in the model, we were able to observe the apparition of the documented morphology of 3D tumour spheroids (Zanoni et al. 2016). The model simulates tumour growth in the absence of the surrounding healthy cells. This assumption was considered in order to reduce the computational cost. We expect that surrounding cells will exert mechanical force which would result in the contraction of the tumour and the minimization the gaps between cells. The present modelling framework can be used to investigate the effects of various biophysical parameters on the growth rate of the tumour such as the size of the cell and diffusion coefficients. We expect that increasing the cell size or decreasing the diffusion coefficients would reduce the occurrence of reactions. As a result, the activation of TFs would diminish which would, in turn, decrease the chances of cell proliferation.

However, the model introduced in this study presents few limitations. The main one concerns the high computational cost due to the complex and physiologically relevant modelling of individual cells. While it is possible to describe the growth of tumours with a low number of cells, large-scale simulations become very expensive for the computational power a normal computer. Therefore, the number of proteins involved in the regulation of cells was reduced in order to decrease the computational cost and make numerical simulations affordable. An important assumption that was considered in this model concerns the shape of receptors. These sites are considered as two clamped spheres, whereas it is probably better to represent them as cylinders. This simplifying hypothesis was considered because of the simplicity in treating spherical geometries. Other simplifications considered in the framework concern the EGFR/ERK pathway. We have captured the essential functioning of this cascade by considering only three proteins: Ras, Raf, ERK, whereas in reality, it contains many more components and feedback mechanisms (Orton et al. 2009; Schoeberl et al. 2002).

Overall, the multiscale modelling framework presented in this work is most appropriate for studies in systems biology. While most multiscale models of tumour growth focus on higher-level physiology and interactions, mutational alterations directly affect the functioning of individual cells. Hence, it is important to accurately capture such effects on the cell-scale level in order to properly simulate the development of tumours. Among the features of the presented framework is the possibility to integrate different scale-specific data extracted from the biomedical literature. In this regard, it is possible to integrate the data obtained by a broad range of acquisition techniques such as flow cytometry, nano-imaging, and transcriptome sequencing. The framework also considers the various perturbations and fluctuations that shape the biological processes at different scales which has the potential to affect the global behaviour of the system. In a forthcoming study, we will apply our framework to assess the efficiency of EGFR inhibitors in the treatment of different types of cancer. In particular, we would like to study how biological noises affect the efficacy of these treatments.

## Notes

### Acknowledgements

This work was supported by the Swedish research council (VR) under Award No. 2015-03964 and by eSSENCE consortium strategic collaboration of eScience.

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