Abstract
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-\(\upkappa \)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-\(\upkappa \)B) and through the use of high-performance computing be capable of simulating up to \(10^9\) cells, i.e. the tissue scale. In this way, mathematical models at multiple scales would be combined to formulate a multiscale computational model.
Similar content being viewed by others
References
Alarcón T, Byrne H, Maini P (2003) A cellular automaton model for tumour growth in inhomogeneous environment. J Theor Biol 225:257–274
Alberts B, Bray D, Hopkin K, Johnson A, Lewis J, Raff M, Roberts K, Walter P (eds) (2010) Essential cell biology. Garland Publishing, Inc., New York
Alcaraz JL, Buscemi M, Grabulosa X, Trepat B, Fabry R, Farre D, Navajas D (2003) Microrheology of human lung epithelial cells measured by atomic force. Biophys J 84:2071–2079
Andasari V, Roper R, Swat MH, Chaplain MAJ (2012) Integrating intracellular dynamics using CompuCell 3D and Bionetsolver: applications to multiscale modelling of cancer cell growth and invasion. PLoS ONE 7(3):e33726
Anderson ARA, Chaplain MAJ (1998) Continuous and discrete mathematical models of tumour-induced angiogenesis. Bull Math Biol 60:857–899
Arenzana-Seisdedos F, Turpin P, Rodriguez M, Thomas D, Hay RT, Virelizier JL, Dargemont C (1997) Nuclear localization of I\(\upkappa \)B alpha promotes active transport of NF-\(\upkappa \)B from the nucleus to the cytoplasm. J Cell Sci 110(Pt 3):369–378
Ashall L, Horton CA, Nelson DE, Paszek P, Harper CV, Sillitoe K, Ryan S, Spiller DG, Unitt JF, Broomhead DS, Kell DB, Rand DA, Sée V, White MRH (2009) Pulsatile stimulation determines timing and specificity of NF-\(\upkappa \)B-dependent transcription. Science 324:242–246
Baker AH, Falgout RD, Kolev TV, Yang UM (2012) Scaling hypre’s multigrid solvers to 100,000 cores. In: Berry MW, Gallivan KA, Gallopoulos E, Grama A, Philippe B, Saad Y, Saied F (eds) High-performance scientific computing. Springer, Berlin, pp 261–279
Balagadde FK, Song H, Ozaki J, Collins CH, Barnet M, Arnold FH, Quake SR, You L (2008) A synthetic Escherichia coli predator–prey ecosystem. Mol Syst Biol 4:187
Bar-On D, Wolter S, van de Linde S, Heilemann M, Nudelman G, Nachliel E, Gutman M, Sauer M, Ashery U (2012) Super-resolution imaging reveals the internal architecture of nano-sized syntaxin clusters. J Bio Chem 287:27158–27167
Barrio M, Burrage K, Leier A, Tian T (2006) Oscillatory regulation of HES1: discrete stochastic delay modelling and simulation. PLoS Comput Biol 2(9):e117
Becskei A, Serrano L (2000) Engineering stability in gene networks by autoregulation. Nature 405:590–593
Bernard S, Čajavec B, Pujo-Menjouet L, Mackey MC, Herzel H (2006) Modeling transcriptional feedback loops: the role of Gro/TLE1 in Hes1 oscillations. Philos Trans A Math Phys Eng Sci 15:1155–1170
Bertuzzi A, Gandolfi A (2000) Cell kinetics in a tumour cord. J Theor Biol 204:587–599
Bertuzzi A, Fasano A, Filidoro L, Gandolfi A, Sinisgalli C (2005) Dynamics of tumour cords following changes in oxygen availability: a model including a delayed exit from quiescence. Math Comput Model 41:1119–1135
Bertuzzi A, Fasano A, Gandolfi A, Sinisgalli C (2010) Necrotic core in EMT6/Ro tumour spheroids: is it caused by an ATP deficit? J Theor Biol 262:142–150
Betzig E, Patterson GH, Sougrat R, Lindwasser OW, Olenych S, Bonifacino JS, Davidson MW, Lippincott-Schwartz J, Hess HF (2006) Imaging intracellular fluorescent proteins at nanometer resolution. Science 313:1642–1645
Busenberg S, Mahaffy JM (1985) Interaction of spatial diffusion and delays in models of genetic control by repression. J Math Biol 22:313–333
Casciari J, Sotirchos S, Sutherland R (1992) Mathematical modelling of microenvironment and growth in EMT6/Ro multicellular tumour spheroids. Cell Prolif 25:1–22
Chaplain MAJ, Ptashnyk M, Sturrock M (2015) Hopf bifurcation in a gene regulatory network model: molecular movement causes oscillations. Math Model Methods Appl Sci 25(6):1179–1215
Chen YY, Galloway KE, Smolke CD (2012) Synthetic biology: advancing biological frontiers by building synthetic systems. Genome Biol 13:240
Cheong R, Hoffmann A, Levchenko A (2008) Understanding NF-\(\upkappa \)B signaling via mathematical modeling. Mol Syst Biol 4:192
Chu Y-S, Thomas WA, Eder O, Pincet E, Thiery JP, Dufour S (2004) Force measurements in E-cadherin-mediated cell doublets reveal rapid adhesion strengthened by actin cytoskeleton remodeling through Rac and Cdc42. J Cell Biol 167:1183–1194
Cytowski M, Szymańska Z (2014) Large scale parallel simulations of 3-D cell colony dynamics. IEEE Comput Sci Eng 16(5):86–95
Cytowski M, Szymańska Z (2015a) Enabling large scale individual-based modelling through high performance computing. In: ITM Web of Conferences, vol 5, p 00014
Cytowski M, Szymańska Z (2015b) Large scale parallel simulations of 3-D cell colony dynamics. II. Coupling with continuous description of cellular environment. IEEE Comput Sci Eng 17(5):44–48
Cytowski M, Szymańska Z, Umiński P, Andrejczuk G, Raszkowski K (2017) Implementation of an agent-based parallel tissue modelling framework for the Intel MIC architecture. Sci Program 2017, Article ID 8721612, 11 pages. doi:10.1155/2017/8721612
D’Antonio G, Macklin P, Preziosi L (2013) An agent-based model for elasto-plastic mechanical interactions between cells, basement membrane and extracellular matrix. Math Biosci Eng 10:75–101
Drasdo D, Höhme S (2005) A single-cell-based model of tumor growth in vitro: monolayers and spheroids. Phys Biol 2:133–147
Drawert B, Engblom S, Hellander A (2012) URDME: a modular framework for stochastic simulation of reaction-transport processes in complex geometries. BMC Syst Biol. doi:10.1186/1752-0509-6-76
Elowitz MB, Leibler S (2000) A synthetic oscillatory network of transcriptional regulators. Nature 403:335–338
Engblom S, Ferm L, Hellander A, Lötstedt P (2009) Simulation of stochastic reaction–diffusion processes on unstructured meshes. SIAM J Sci Comput 31:1774–1797
Galle J, Loeffler M, Drasdo D (2005) Modelling the effect of deregulated proliferation and apoptosis on the growth dynamics of epithelial cell populations in vitro. Biophys J 88:62–75
Geva-Zatorsky N, Rosenfeld N, Itzkovitz S, Milo R, Sigal A, Dekel E, Yarnitzky T, Liron Y, Polak P, Lahav G, Alon U (2006) Oscillations and variability in the p53 system. Mol Syst Biol. doi:10.1038/msb4100068
Gibson MA, Bruck J (2000) Efficient exact stochastic simulation of chemical species and many channels. J Phys Chem 104:1876–1889
Gillespie DT (1976) A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J Comput Phys 22(4):403–434
Glass L, Kauffman SA (1970) Co-operative components, spatial localization and oscillatory cellular dynamics. J Theor Biol 34:219–237
Goodwin BC (1965) Oscillatory behaviour in enzymatic control processes. Adv Enzyme Regul 3:425–428
Griffith JS (1968) Mathematics of cellular control processes. I. Negative feedback to one gene. J Theor Biol 20:202–208
Gumbiner BM (2005) Regulation of cadherin-mediated adhesion in morphogenesis. Nat Rev Mol Cell Biol 6:622–634
Hanahan D, Weinberg RA (2000) The hallmarks of cancer. Cell 100:57–70
Hanahan D, Weinberg RA (2011) Hallmarks of cancer: the next generation. Cell 144:646–674
Harang R, Bonnet G, Petzold LR (2012) WAVOS: a MATLAB toolkit for wavelet analysis and visualization of oscillatory systems. BMC Res Notes 5:163
Hiersemenzel K, Brown ER, Duncan RR (2013) Imaging large cohorts of single ion channels and their activity. Front Endocrinol. doi:10.3389/fendo.2013.00114
Hirata H, Yoshiura S, Ohtsuka T, Bessho Y, Harada T, Yoshikawa K, Kageyama R (2002) Oscillatory expression of the bHLH factor Hes1 regulated by a negative feedback loop. Science 298:840–843
Hlatky L, Hahnfeldt P, Folkman J (2002) Clinical application of anti-angiogenic therapy: microvessel density, what it does and doesn’t tell us. J Natl Cancer Inst 94(12):883–893
Hoffmann A, Levchenko A, Scott M, Baltimore D (2002) The I\(\upkappa \)B–NF-\(\upkappa \)B signaling module: temporal control and selective gene activation. Science 298:1241–1245
Jagiella N, Müller B, Müller M, Vignon-Clementel IE, Drasdo D (2016) Inferring growth control mechanisms in growing multi-cellular spheroids of nsclc cells from spatial-temporal image data. PLoS Comput Biol 12(2):e1004412
Jensen MH, Sneppen J, Tiana G (2003) Sustained oscillations and time delays in gene expression of protein Hes1. FEBS Lett 541:176–177
Lachowicz M, Parisot M, Szymańska Z (2016) Intracellular protein dynamics as a mathematical problem. Discrete Contin Dyn Syst B 21:2551–2566
Lahav G, Rosenfeld N, Sigal A, Geva-Zatorsky N, Levine AJ, Elowitz MB, Alon U (2004) Dynamics of the p53–Mdm2 feedback loop in individual cells. Nature Genet 36:147–150
Lee RE, Walker SR, Savery K, Frank DA, Gaudet S (2014) Fold change of nuclear NF-\(\upkappa \)B determines TNF-induced transcription in single cells. Mol Cell 53(6):867–879
Lewis J (2003) Autoinhibition with transcriptional delay: a simple mechanism for the zebrafish somitogenesis oscillator. Curr Biol 13:1398–1408
Lipniacki T, Kimmel M (2007) Deterministic and stochastic models of NF\(\upkappa \)B pathway. Cardiovasc Toxicol 7:215–234
Mackey MC, Glass L (1977) Oscillation and chaos in physiological control systems. Science 197:287–289
Macklin P, McDougall S, Anderson ARA, Chaplain MAJ, Cristini V, Lowengrub J (2009) Multiscale modelling and nonlinear simulation of vascular tumour growth. J Math Biol 58:765–798
Macnamara CK, Chaplain MAJ (2016) Diffusion driven oscillations in gene regulatory networks. J Theor Biol 407:51–70
Macnamara CK, Chaplain MAJ (2017) Spatio-temporal models of synthetic genetic oscillators. Math Biol Eng 14:249–262
Mahaffy JM (1988) Genetic control models with diffusion and delays. Math Biosci 90:519–533
Mahaffy JM, Pao CV (1984) Models of genetic control by repression with time delays and spatial effects. J Math Biol 20:39–57
Mahaffy RE, Shih CK, McKintosh FC, Kaes J (2000) Scanning probe-based frequency-dependent microrheology of polymer gels and biological cells. Phys Rev Lett 85:880–883
Manley S, Gillette JM, Patterson GH, Shroff H, Hess HF, Betzig E, Lippincott-Schwartz J (2008) High-density mapping of single-molecule trajectories with photoactivated localization microscopy. Nat Methods 5:155–157
Marquez-Lago TT, Leier A, Burrage K (2010) Probability distributed time delays: integrating spatial effects into temporal models. BMC Syst Biol. doi:10.1186/1752-0509-4-19
McDougall SR, Anderson ARA, Chaplain MAJ (2006) Mathematical modelling of dynamic adaptive tumour-induced angiogenesis: clinical implications and therapeutic targeting strategies. J Theor Biol 241:564–589
Miron-Mendoza M, Koppaka V, Zhou C, Petroll WM (2013) Techniques for assessing 3-D cellmatrix mechanical interactions in vitro and in vivo. Exp Cell Res 319:2470–2480
Momiji H, Monk NAM (2008) Dissecting the dynamics of the Hes1 genetic oscillator. J Theor Biol 254:784–798
Monk NAM (2003) Oscillatory expression of Hes1, p53, and NF-\(\upkappa \)B driven by transcriptional time delays. Curr Biol 13:1409–1413
Mueller-Klieser WF, Sutherland RM (1984) Oxygen consumption and oxygen diffusion properties of multicellular spheroids from two different cell lines. Adv Exp Med Biol 180:311–321
Näthke IS, Hinck L, Nelson WJ (1995) The cadherin/catenin complex: connections to multiple cellular processes involved in cell adhesion, proliferation and morphogenesis. Semin Dev Biol 6:89–95
Nelson DE, Ihekwaba AEC, Elliott M, Johnson JR, Gibney CA, Foreman BE, Nelson G, See V, Horton CA, Spiller DG, Edwards SW, McDowell HP, Unitt JF, Sullivan E, Grimley R, Benson N, Broomhead D, Kell DB, White MRH (2004) Oscillations in NF-\(\upkappa \)B signaling control the dynamics of gene expression. Science 306:704–708
O’Brien EL, Itallie EV, Bennett MR (2012) Modeling synthetic gene oscillators. Math Biosci 236:1–15
O’Dea E, Hoffmann A (2010) The regulatory logic of the NF-\(\upkappa \)B signaling system. Cold Spring Harb Perspect Biol 2(1):a00021
Pekalski J, Zuk P, Kochanczyk M, Junkin M, Kellogg R, Tay S, Lipniacki T (2013) Spontaneous NF\(\upkappa \)B activation by autocrine TNF\(\upalpha \) signaling: a computational analysis. PLoS ONE 8(11):e78887
Purcell O, Savery NJ, Grierson CS, di Bernardo M (2010) A comparative analysis of synthetic genetic oscillators. J R Soc Interface 7:1503–1524
Ramis-Conde I, Drasdo D, Anderson ARA, Chaplain MAJ (2008) Modelling the influence of the E-cadherin-\(\upbeta \)-catenin pathway in cancer cell invasion: a multi-scale approach. Biophys J 95:155–165
Ramis-Conde I, Drasdo D, Anderson ARA, Chaplain MAJ (2009) Multi-scale modelling of cancer cell intravasation: the role of cadherins in metastasis. Phys Biol 6:016008
Ritchie T, Zhou W, McKinstry E, Hosch M, Zhang Y, Näthke IS, Engelhardt JF (2001) Developmental expression of catenins and associated proteins during submucosal gland morphogenesis in the airway. Exp Lung Res 27:121–141
Schaller G, Meyer-Hermann M (2005) Multicellular tumor spheroid in an off-lattice Voronoi–Delaunay cell model. Phys Rev E 71:051910-1–051910-16
Schlüter DK, Ramis-Conde I, Chaplain MAJ (2012) Computational modeling of single cell migration: the leading role of extracellular matrix fibers. Biophys J 103:1141–1151
Schlüter DK, Ramis-Conde I, Chaplain MAJ (2015) Multi-scale modelling of the dynamics of cell colonies: insights into cell-adhesion forces and cancer invasion from in silico simulations. J R Soc Interface 12:20141080
Shirinifard A, Gens J, Zaitlen B, Poplawski N, Swat M, Glazier J (2009) 3D multi-cell simulation of tumor growth and angiogenesis. PLoS ONE 4(10):e7190
Shymko RM, Glass L (1974) Spatial switching in chemical reactions with heterogeneous catalysis. J Chem Phys 60:835–841
Skaug B, Chen J, Du F, He J, Ma A, Chen ZJ (2011) Direct, noncatalytic mechanism of IKK inhibition by A20. Mol Cell 44(4):559–571
Smolen P, Baxter DA, Byrne JH (1999) Effects of macromolecular transport and stochastic fluctuations on the dynamics of genetic regulatory systems. Am J Physiol 277:C777–C790
Smolen P, Baxter DA, Byrne JH (2001) Modeling circadian oscillations with interlocking positive and negative feedback loops. J Neurosci 21:6644–6656
Smolen P, Baxter DA, Byrne JH (2002) A reduced model clarifies the role of feedback loops and time delays in the Drosophila circadian oscillator. Biophys J 83:2349–2359
Spiller DG, Wood CD, Rand DA, White MRH (2010) Measurement of single-cell dynamics. Nature 465(7299):736–45
Sturrock M, Terry AJ, Xirodimas DP, Thompson AM, Chaplain MAJ (2011) Spatio-temporal modelling of the Hes1 and p53–Mdm2 intracellular signalling pathways. J Theor Biol 273:15–31
Sturrock M, Terry AJ, Xirodimas DP, Thompson AM, Chaplain MAJ (2012) Influence of the nuclear membrane, active transport, and cell shape on the Hes1 and p53–Mdm2 pathways: insights from spatio-temporal modelling. Bull Math Biol 74:1531–1579
Sturrock M, Hellander A, Matzavinos A, Chaplain MAJ (2013) Spatial stochastic modelling of the Hes1 gene regulatory network: intrinsic noise can explain heterogeneity in embryonic stem cell differentiation. J R Soc Interface 10:20120988
Szymańska Z, Parisot M, Lachowicz M (2014) Mathematical modeling of the intracellular protein dynamics: the importance of active transport along microtubules. J Theor Biol 363:118–128
Thompson DW (1917) On growth and form. Cambridge University Press, Cambridge
Tian T, Burrage K, Burrage PM, Carlettib M (2007) Stochastic delay differential equations for genetic regulatory networks. J Comput Appl Math 205(2):696–707
Tiana G, Jensen MH, Sneppen K (2002) Time delay as a key to apoptosis induction in the p53 network. Eur Phys J B 29:135–140
van de Linde S, Löschberger A, Klein T, Heidbreder M, Wolter S, Heilemann M, Sauer M (2011) Direct stochastic optical reconstruction microscopy with standard fluorescent probes. Nat Protoc 6:991–1009
Walenta S, Mueller-Klieser WF (1987) Oxygen consumption rate of tumour cells as a function of their proliferative status. Adv Exp Med Biol 215:389–391
Weinberg RA (2007) The biology of cancer. Garland Science, New York
Won S, Lee B-C, Park C-S (2011) Functional effects of cytoskeletal components on the lateral movement of individual BK Ca channels expressed in live COS-7 cell membrane. FEBS Lett 585:2323–2330
Yordanov B, Dalchau N, Grant PK, Pedersen M, Emmott S, Haseloff J, Phillips A (2014) A computational method for automated characterization of genetic components. ACS Synth Biol 3:578–588
Zacharaki E, Stamatakos G, Nikita K, Uzunoglu N (2004) Simulating growth dynamics and radiation response of avascular tumour spheroids: model validation in the case of an EMT6/Ro multicellular spheroid. Comput Methods Programs Biomed 76:193–206
Zaman MH, Trapani LM, Sieminski AL, MacKellar D, Gong H, Kamm RD, Wells A, Lauffenburger DA, Matsudaira P (2006) Migration of tumor cells in 3D matrices is governed by matrix stiffness along with cell–matrix adhesion and proteolysis. Proc Natl Acad Sci 103:10889–10894
Acknowledgements
ZS acknowledges the support of the National Science Centre Poland Grant 2011/01/D/ST1/04133, National Science Centre Poland Grant 2014/15/B/ST6/05082 and The National Centre for Research and Development Grant STRATEGMED1/233224/10/NCBR/2014. MAJC and CKM gratefully acknowledge support of EPSRC Grant No. EP/N014642/1 (EPSRC Centre for Multiscale Soft Tissue Mechanics—With Application to Heart & Cancer). EM was supported by an EASTBIO Ph.D. Fellowship. The authors thank Bartosz Borucki from ICM for his help in with the VisNow medical imaging software.
Author information
Authors and Affiliations
Corresponding author
Appendices
Appendix 1: The Reaction–Diffusion Master Equation
Here we describe the formulation of the spatial stochastic model for intracellular GRN dynamics. We describe the computational domain and how reaction and diffusion events are incorporated into the reaction-diffusion master equation (RDME). We also provide some notes on how simulations are produced. For a more detailed description, see the supplementary material of Sturrock et al. (2013).
1.1 The Computational Domain
The computational domain (see Fig. 2, for example) is set up using COMSOL and a mesh is imposed. In general, the domain \(\varOmega \) is meshed into V tetrahedra-shaped subvolumes, voxels, \(\varOmega _k, k \in \{1,\ldots ,V\}\) such that,
At any given time, the state of the system is described by the number of each chemical species within the domain. Changes to the state will either be by the chemical reactions at the voxel level or the movement (diffusion jumps) of a molecule between neighbouring voxels.
1.2 Chemical Reactions
We consider reactions that occur due to molecular contact. We assume that the species of our system, within each subvolume, are uniformly distributed and in thermal equilibrium, such that the motion of each molecule is random. We consider the probability of a collision occurring between two reactant molecules. The likelihood of a reaction occurring, changing the state of the system from x to \(x+N_r\), is determined by its reaction rate, described by the reaction propensity function \(\omega _r(x)\). As such, reactions can be described by
where \(N_r \in \mathbb {Z}^S\) is the transition step, defined by the rth column of the stoichiometric matrix M and \(\omega _r(x)\) is the probability that the reaction occurs during a infinitesimal time interval, i.e.
1.3 Molecular Diffusion
The movement of a chemical species \(S_l\) from a voxel \(\psi _i\) to a randomly selected adjacent voxel \(\psi _j\) describes the molecular diffusion and is modelled as a first-order event. As such, we treat the diffusive process in a similar way to a reactive process and consider the probability of a transition taking place, i.e. the probability for one of the lth species to make a jump from the ith subvolume to an adjacent jth subvolume. Hence, we consider the following,
where \(x_{{li}}\) is number of species l located in voxel i and \(q_{lij}\) is the diffusion rate constant that depends on the macroscopic diffusion coefficient of species l (\(D_l\)) and the mesh of the domain, specifically the shape and size of voxels \(\psi _i\) and \(\psi _j\). Note each \(q_{lij}\) is only non-zero for connected mesh elements and of the types of species we model only mRNAs and proteins diffuse, the free and occupied promoters remain permanently within the voxel assigned as the promoter site.
1.4 Solving the System
The temporal evolution of the probability distribution of each state in the state space is governed by the RDME. We complete the model set-up with zero-flux boundary conditions at the cell membrane, while we impose continuity of flux on the nuclear membrane. For initialisation, we suppose that there is only a single free promoter within each promoter site. All simulations found in this paper are produced using the URDME (Unstructured Reaction-Diffusion Master Equation) software framework (Drawert et al. 2012), which implements the next subvolume method (NSM) (Gibson and Bruck 2000); the NSM being far more computationally efficient than the classical SSA for a 3D domain such as ours. URDME uses unstructured tetrahedral and triangular meshes (such as shown in Fig. 2) for which diffusion rate constants \(q_{lij}\) are automatically computed (Engblom et al. 2009; Drawert et al. 2012).
Appendix 2: NF-\(\upkappa \)B Reactions
Here we give the reactions for the NF-\(\upkappa \)B pathway (Tables 6, 7, 8).
Rights and permissions
About this article
Cite this article
Szymańska, Z., Cytowski, M., Mitchell, E. et al. Computational Modelling of Cancer Development and Growth: Modelling at Multiple Scales and Multiscale Modelling. Bull Math Biol 80, 1366–1403 (2018). https://doi.org/10.1007/s11538-017-0292-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-017-0292-3
Keywords
- Multiscale cancer modelling
- Gene regulatory network
- Spatial stochastic model
- Individual-based model
- Computational simulations