In this paper, we present two mathematical models related to different aspects and scales of cancer growth. The first model is a stochastic spatiotemporal model of both a synthetic gene regulatory network (the example of a three-gene repressilator is given) and an actual gene regulatory network, the NF-[Formula: see text]B pathway. The second model is a force-based individual-based model of the development of a solid avascular tumour with specific application to tumour cords, i.e. a mass of cancer cells growing around a central blood vessel. In each case, we compare our computational simulation results with experimental data. In the final discussion section, we outline how to take the work forward through the development of a multiscale model focussed at the cell level. This would incorporate key intracellular signalling pathways associated with cancer within each cell (e.g. p53-Mdm2, NF-[Formula: see text]B) and through the use of high-performance computing be capable of simulating up to [Formula: see text] cells, i.e. the tissue scale. In this way, mathematical models at multiple scales would be combined to formulate a multiscale computational model.

Transcription factors are important molecules which control the levels of mRNA and proteins within cells by modulating the process of transcription (the mechanism by which mRNA is produced within cells) and hence translation (the mechanism by which proteins are produced within cells).Transcription factors are part of a wider family of molecular interaction networks known as gene regulatory networks (GRNs) which play an important role in key cellular processes such as cell division and apoptosis (e.g. the p53-Mdm2, NFκB pathways). Transcription factors exert control over molecular levels through feedback mechanisms, with proteins binding to gene sites in the nucleus and either up-regulating or down-regulating production of mRNA. In many GRNs, there is a negative feedback in the network and the transcription rate is reduced. Typically, this leads to the mRNA and protein * Corresponding author. Phone: +44 (0)1334 463723Email addresses: ckm@st-andrews.ac.uk (Cicely K Macnamara), emitchell@maths.dundee.ac.uk (Elaine I Mitchell), majc@st-andrews.ac.uk (Mark AJ Chaplain)Author accepted manuscript 56 to capture the effects of stochasticity in a single cell. In this paper, then, we 57 consider the more biologically relevant discrete, spatial-stochastic approach 58 derived from the spatial-stochastic model of the Hes1 GRN put forward by 59 Sturrock et al. (2013). PDE models for repressilators and activator-inhibitors 60 showed that oscillations may be achieved provided the relationship between 61 the spatial location of the gene site and diffusion coefficient is optimised 62 (Macnamara and Chaplain, 2016). We will investigate similar themes here, 63 4 discussing how spatio-temporal dynamics change as we vary the location of 64 the gene site(s) and the diffusion coefficient of the mRNA and protein spe-65 cies. Note the term repressilator (introduced by Elowitz and Leibler, 2000) 66has historically been reserved for a system of three genes which couple to 67 form a cycle of negative feedback, however, for ease of reference we choose 68 to use this terminology, for any n-gene system for which the protein of any 69 given gene inhibits the production of the mRNA for the subsequent gene. Ac-70 cording to our terminology activator-repressor systems couple positive and 71 negative feedback. 72The paper is structured as follows. In Section 2 we layout the specific 73 model(s) to be investigated and give details of how simulations are carried 74 out. In Section 3 we provide results for repressilator systems; first revisiting 75 the Hes1 system, or one-gene repressilator, (as detailed by Sturrock et al., 76 2013) to discuss how changes to spatial aspects affect the molecular dynamics 77 and then extending the approach to a two-gene repressilator system. In 78 Section 4 we present simulation results for a two-gene activator-repressor 79 system which contains both positive and negative feedback. Discussions, 80 conclusions and directions for future work in this area are given in Section 5. 81 2. Model 82 Throughout this paper we invest...